Modeling the physics of RRAM defects

Modeling the physics of RRAMdefects

A model simulating RRAMdefects on a macroscopicphysical level

T. Hol

Modeling thephysics of

RRAM defectsA model simulating RRAMdefects on a macroscopic

physical levelby

T. Holto obtain the degree of Master of Science

at the Delft University of Technology,to be defended publicly on Friday August 28, 2020 at 13:00.

Student number: 4295528Project duration: September 1, 2019 – August 28, 2020Thesis committee: Prof. dr. ir. S. Hamdioui TU Delft, supervisor

Dr. Ir. R. Ishihara TU DelftDr. Ir. S. Vollebregt TU DelftDr. Ir. M. Taouil TU DelftIr. M. Fieback TU Delft, supervisor

An electronic version of this thesis is available at https://repository.tudelft.nl/.

https://repository.tudelft.nl/

Abstract

Resistive RAM, or RRAM, is one of the emerging non-volatile memory (NVM) technologies, which could beused in the near future to fill the gap in the memory hierarchy between dynamic RAM (DRAM) and Flash, oreven completely replace Flash. RRAM operates faster than Flash, but is still non-volatile, which enables it tobe used in a dense 3D NVM array. It is also a suitable candidate for computation-in-memory, neuromorphiccomputing and reconfigurable computing.

However, the show stopping problem of RRAM is that it suffers from unique defects, which is the reasonwhy RRAM is still not widely commercially adopted. These defects differ from those that appear in CMOStechnology, due to the arbitrary nature of the forming process. They can not be detected by conventionaltests and cause defective devices to go unnoticed. Therefore, new tests need to be developed that properlyinclude the physics of a defective device in a RRAM model.

Device-aware testing (DAT) is the state-of-the-art solution to this problem. By accounting for the uniquephysics of an RRAM device, DAT is able to detect unique RRAM defects. However, DAT bases its results onrelatively compact electrical models, which do not account for randomness present in e.g. the forming of thefilament and local temperature fluctuations. Meanwhile, many low-level physical models exist already thatcan model this randomness and provide accurate insights into the physical specifics of RRAM. These modelsdo, however, hardly ever analyze the effects of defects.

The contribution of this work is to expand and improve one of the state-of-the-art physical models toanalyse manufacturing defects on a low, near atomic-level scale. For the first time, the characteristics ofa defect can be described in the physical shape of the defect, rather than only the electrical consequencesof a black box device. This enables deep level analysis and characterization of defects, the results of whichimproves DAT to detect even more unique defects.

The model is applied to four types of RRAM-related defects: oxygen vacancy density fluctuation, oxidethickness variation, electrode roughness, and contamination by impurities. The effect of the defects on theconductivity of the device are observed and explained, and their unique non-linear behavior is confirmed bysimulation.

Finally, a discussion is presented which criticizes the reproducability of the referenced defect-free model,but also shows the potential of this work’s model to be improved. Dynamic defects are not yet included, butthe model does provide a static characterization of unique RRAM defects, improving the quality of DAT.

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Preface

After a year of collecting information, understanding theoretical physics, but mostly attempting differentimplementations, this thesis is finally finished. The great majority of the time spent on this work was tryingagain and again to generate realistic results, taking advice from so many sources and great people - and thenstill, failing again and again. Thus, eventually a decision was made to report the parts of the model thatproduced the most realistic results and draw a conclusion.

Furthermore, the better half of the thesis process had to be done at home, due to the currently still activeglobal pandemic. This definitely did not aid productivity and contributed to the unfortunately unfinishednature of this work.

Nevertheless, it is now done, and I have a lot of people to thank for keeping up with me through the lastyear: first of all, my supervisor Moritz, who was always present to provide me with valuable feedback, help,pointers, and papers. Then, my thesis advisor Said Hamdioui, for always keeping my eyes on the impor-tant parts. Thanks to Ryoichi Ishihara, who helped me understand the concept of phonons and tunneling,and getting me in touch. Also, many thanks to Gennadi Bersuker and Luca Larcher, who, even though theydidn’t have the required information, still helped me as much as they could. Finally, I thank my parents fornever giving up on me, even though I often wanted to, and all my friends at the Delft choir and orchestra forproviding me with the music everyone needs in life.

T. HolDelft, August 2020

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Contents

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Contribution of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Background 52.1 Memory in computing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Emerging memory technologies and RRAM design . . . . . . . . . . . . . . . . . . . . . . . . 72.3 RRAM manufacturing process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 State of the art 173.1 RRAM defect modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Classification of defect-free RRAM models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Macroscopic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.4 Comparison and defect-free model choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Model implementation 254.1 Overview of the model structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 General modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Solving the discrete Poisson equation in 3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Calculating the current: the charge transport model . . . . . . . . . . . . . . . . . . . . . . . 344.5 Determining state change: the kMC engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.6 Injection of defects in the model device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Validation of model 595.1 Validating the solution of the 3D Poisson equation solver . . . . . . . . . . . . . . . . . . . . . 595.2 Validating the calculated device current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.3 Validating the events that change the model state. . . . . . . . . . . . . . . . . . . . . . . . . 645.4 Validating the full model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6 Defect analysis 796.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7 Discussion 87

8 Conclusion 89

Bibliography 91

A Model default properties 99

vii

1Introduction

1.1. MotivationAlmost 50 years ago, Chua theorized the existence of the memristor as the fourth fundamental electroniccomponent next to resistors, capacitors and inductors [24]. It was considered to be just a theoretical ex-pansion of the already existing passive elements, until not too long ago HP Labs claimed to be the first tocreate a device that exhibits the characteristics of a memristor [86]. Since then, the applications of memris-tors, named “memristive devices”, have been explored extensively [85, 95, 101]. These applications includecomputation-in-memory [34, 60, 104], synaptic neuromorphic computing for neural networks [23, 49], re-configurable computing [25, 107] and reconfigurable physical unclonable functions [17]. One of the most ex-tensively explored applications of memristive devices, however, is the implementation as Non-Volatile Mem-ory (NVM) [53, 64, 68, 101].

Figure 1.1: A schematic representation of the layout of a memristive crossbar memory [101].

Dense crossbar structures of these devices [53] are speculated to be able to contend with existing 3DNAND flash, given their comparable feature size, but faster read and write times [68, 101]. A schematicrepresentation of a crossbar memory is displayed in Figure 1.1. This layout makes the devices usable asnon-volatile Random Access Memory (RAM). The most prominent three of these technologies [101] are Spin-Transfer-Torque Magnetic RAM (STT-MRAM) [52], Phase-Change RAM (PCRAM) [98] and Resistive RAM (ReRAMor RRAM)[97]. The first two exhibit resistance switching, but are not true memristors. However, the last one,RRAM, behaves like a memristor and can therefore also be used for applications apart from NVM as citedbefore.

RRAM is based on the formation and rupture of a conductive filament through an oxide, a mechanismwhich was known for a long time already [45] and was also the mechanism behind HP’s first memristive de-vice [86]. A notable advantage of RRAM over other emerging NVM technologies is the fact that it can be

1

2 1. Introduction

integrated in the back end of line (BEOL) process of existing CMOS technology [97] and is therefore the lead-ing candidate for large-scale commercial use. The manufacturing process of RRAM involves a controlleddielectric breakdown of an oxide [97]. Applying a reverse bias to this filament reoxidizes part of the filament,whereas applying a forward bias recreates the dielectric breakdown in the oxidized part. This changes theconductive properties of this device and allows it to switch between a High Resistance State (HRS) and a LowResistance State (LRS). Unfortunately, the formation of a filament occurs randomly and abruptly, causingmuch variability in the resistances of the two states of the device [19, 29, 79, 81, 105]. Mitigation tactics havebeen invented to deal with the variability by making use of clever forming techniques [41, 65, 66]. However,these forming techniques are not perfect. Advanced forming techniques can effectively narrow the spread ofHRS resistances, but are often too slow and complex to be applied on a large scale in a commercial context.

Another issue with the manufacturing of RRAM is that variations in the manufacturing process can alsocreate defects. The existing fault models, strategies of describing the effect of defects, have previously sufficedto check for defective devices [70]. Inserting linear elements such as resistors in series or in parallel to a device[5] can model the effects of shorts or opens caused by process-induced defects. However, RRAM defects haveunique effects because the formation of a filament can fail in unexpected ways [93]. Therefore, new faultmodels need to be developed. Examples are the undefined state (US) fault [30, 44], where it is unclear whethera cell carries a 1 or a 0, or deep state (DS) fault [50], where a too thick or too thin filament causes a device tobe unable to transition under regular applied write voltages. It has been proven that these defects can notbe modelled by simple linear elements [31, 32]: since an RRAM behaves like a memristor [97], its resistancechanges according to its applied voltage, the duration is was applied, and the previous shape of the filament.The lack of in-depth knowledge about these kinds of defects causes them to go unnoticed, resulting in testescapes [30].

1.2. State of the ArtIn the state of the art, steps have already been made to model the defects in RRAM devices, not by linearelements, but on a device-aware level, with the goal to develop more effective tests. Fieback et al. propose theprinciple of Device-Aware Testing (DAT), where the physical characteristics of a device defect are observedand incorporated in a dedicated defect model. In this way, DAT-models look beyond linear characterizationsof defects and introduce non-linear alternatives. These alternatives are based mostly on observations on anelectrical level, but lack a general connection to the actual physical processes that occur on a molecular level.Such a physical model is necessary to explain the non-linear characteristics that DAT-models use.

Fortunately, many models already exist that simulate the forming and resistive switching process of RRAMdevices [57], some attempting to model atomic physical RRAM processes [27, 28, 109], some connecting theseprocesses to electrical characteristics [1, 43, 69, 76, 81, 82] and some connecting these electrical character-istics to a larger context [20, 42, 62, 84], such as a memory array of several devices [62], or neuromorphiccomputing [23]. In short, these three levels of model abstraction can be called microscopic, macroscopic andcompact [57]. With a few exceptions [1, 16, 31], all of these models assume a perfectly manufactured RRAMdevice.

Microscopic models are too complex for simulating the full resistive switching process, but provide valu-able information for macroscopic models. On the other hand, compact models can easily simulate multipleswitching cycles, but lack a direct connection to the atomic physics of microscopic models. Existing RRAMmodels that incorporate manufacturing defects and process variations [1, 16, 30] do successfully reproducethe electrical behavior of defective devices, but do not explain the reason of this behavior. Thus, a macro-scopic, general defect model is needed, that can provide the necessary connection between electrical char-acteristics of compact models and physical processes of microscopic models.

1.3. Contribution of this workThis work fills the gap between RRAM physics and RRAM electrical behavior, in the case of a defective de-vice. It connects electrical characteristics, that can be measured, to the underlying physical processes, thatare more difficult to observe. The model combines a detailed analysis of these physical processes with anoptimized MATLAB framework. Thus it gives a thorough, physics-based insight on the effects of defects andprocess variations on the operation of RRAM devices.In short, the contribution of this work consists of:

• A complete analysis of the most relevant RRAM physical processes that enable resistive switching.

1.4. Outline 3

• An overview of the common RRAM manufacturing process, the defects that occur in it, and a descrip-tion of these defects to connect them to a physical model.

• A physical RRAM model that connects the defects to electrical device characteristics to accurately re-produce the behavior of defective RRAM devices.

• An open-source MATLAB framework implementing the model that provides a large range of parametersand properties to modify and automatically sweep over.

• A validation and subsequent argumentation of the physical model, criticizing its viability and reprodu-cability.

• An insight in the effects of oxygen thickness variation, electrode roughness and arbitrarily-shaped im-purities, on the electrical characteristics of RRAM devices backed by physics.

1.4. OutlineThe remainder of this thesis is organised as follows. First, background information, that explains all the basicsnecessary for understanding the specifics of the model implementation, will be provided in Chapter 2. Thenin Chapter 3, existing research into RRAM modeling - with and without inclusion of defects and processvariation - is explored. In Chapter 4, an in-depth explanation of this work’s model implementation is given.Chapter 5 validates parts of the model and provides argumentation to explain the invalidated parts. Afterthat, in Chapter 6 the model is put to use to observe the effect of manufacturing defects. Finally, this work isconcluded with a discussion in Chapter 7 and a conclusion in Chapter 8.

2Background

This chapter will provide the background information that is required to understand the relevance of mem-ory in computing, emerging memory technologies and RRAM modeling. First, the general context of memoryin computing is illustrated. Then, the principles of emerging non-volatile memory (NVM) technologies andRRAM design and operation are outlined, together with its opportunities and challenges in the implemen-tation as a memory array. After that an insight is given in the manufacturing process of RRAM. The chapterconcludes with an analysis of defects that may occur during the manufacturing process of an RRAM devices.

2.1. Memory in computingBefore considering the specifics of RRAM, it is a good idea to take a step back and consider the context. First,we look at the purpose and position of memory in a computing system. Then the problem of the memorygap is considered, followed by its mitigation strategy, memory hierarchy. Eventually an overview is given ofexisting, present day memory technologies.

On a broad scale, a computing system can be summarized into four parts [26]:

Figure 2.1: The four basic parts of a computer system: input, output, Processor/CPU and memory [26].

Figure 2.1 provides a complete overview of these four basic parts.

• Input: a user provides instructions and information to the computing system, telling it what it needsto do.

• Output: after processing the provided information, the system returns results to the user.

• Processor or CPU: (central processing unit) acts as the controller of the full system, performing opera-tions on the provided information.

5

6 2. Background

• Memory: saves the provided and the to be returned information, as well as intermediary results duringprocessing.

The processor receives instructions from the user in assembly code via a compiler. This is why the pro-cessor is considered the most important part of a computer, but it cannot function without a memory. Thememory is used as a storage for program instructions and intermediate data. Therefore, it gets accessed fre-quently during operation of the processor. The processor is, however, historically [96, 100] faster than thememory, an issue that is called the Memory Wall or the Memory Gap.

The Memory GapIn 1965 Gordon E. Moore, now chairman emeritus of Intel Corp., quantified the rapid growth of semiconduc-tor technology by an exponential function that is now known as Moore’s Law [83]. His prediction became aself-fulfilled prophecy as processors became continuously faster and smaller: every two years, their speed in-creased by at least 50%, thanks to technology scaling and strategies such as instruction level parallelization.However, as computing power grew exponentially, the speed of memory could not keep up because muchless speedup strategies were invented for memory. This caused the issue called the Memory Gap [96], namedafter the still growing gap between memory and processing speed. In modern processors, storing and loadingdata is still the bottleneck in computer operation, and by far the largest area of it is covered in memory. Thememory gap can however be mitigated by making use of several different types of memory, that have opti-mized properties for different uses. When these memories are sorted according to their optimizations, theyform the memory hierarchy.

Memory hierarchyFor a software programmer, the ideal memory is infinitely large and has instant access to requested data. Ofcourse, realistically, memory size and access time have limits, but by exploiting the strengths of the differenttypes of memory and data locality, these limits can be stretched. Data locality is the principle of data existingclose to each other for computing purposes, either in time (temporal locality) or in space (spatial locality)[26]. If it is not necessary to keep data close, it can be stored further away from the CPU and retrieved whenit is needed.

Multiple levels can be defined in this way, where, moving further away from the CPU, data is accessed lessoften and speed is less of an issue, which means that the level can be focused more on storage capacity. Thissystem of levels is called memory hierarchy. An example of memory hierarchy in a computer system is givenin Figure 2.2. Registers and caches are implemented closer to the processor because of their quick accesstimes. A level below that, DRAM is used to store more data in the Random Access Memory which needs to beaccessed less frequently. Then, at the base there is the main memory, a non-volatile, large data storage. Themain memory uses Flash technology or disk technology.

If the processor needs a certain piece of information, it asks the cache. If the cache does not have theinformation (a cache miss), the cache asks the RAM for the data. If the RAM does not have the information, itasks the flash memory, which then in turn asks the main memory. The data moves up along the hierarchy andthe relevant information is stored in the cache. In this way, the bottom of the hierarchy can be optimized foran affordable, large capacity, whereas the top of the hierarchy is optimized for speed and accessibility. Theseoptimizations are a consequence of the technologies used for the types of memory in the levels.

Types of memoryThere are numerous methods of storing data on a chip, but in the last few decades, the most common tech-nologies have been SRAM, DRAM, Flash and magnetic storage [3, 11]. These technologies range from fast butexpensive, to cheap but slow. Another trait of memory is volatility: if the power to a memory cell is removed,volatile memory will lose its state, while non-volatile memory (NVM) will retain its state. A list of the fourmemory technologies follows below:

• SRAM or Static Random Access Memory (SRAM) is implemented as two cross-coupled inverters withan additional two transistors acting as selectors, using a total of 6 transistors [3]. SRAM exists as theregisters and the cache on the central processing unit. It is the fastest, but also most expensive memoryimplementation. SRAM is volatile memory.

• DRAM or Dynamic Random Access Memory uses just a single transistor and a small capacitor to storecharge. It is smaller and thus cheaper per unit than SRAM, but the charging of the DRAM cell takes more

2.2. Emerging memory technologies and RRAM design 7

Figure 2.2: The five different levels of memory hierarchy in a computer system [26].

time than setting an SRAM [3]. DRAM exists in separate units in a computer system and is colloquiallyreferred to as just "Random Access Memory" or RAM. DRAM is also volatile.

• Flash uses a single transistor and saves data by charging a transistor gate surrounded by dielectric, orfloating gate in order to change its threshold voltage [11]. It is slower than DRAM, but has the addedbenefit of being non-volatile. In the past, flash was mostly used in small external memories such asUSB storages and digital camera’s [11], but today the technology is also present in solid state disks(SSDs) [110].

• Magnetic storage is the oldest method of storing data, first developed in the 1950s [3]. It uses magneticfields, read and written on a spinning disk by a moving head, or on a moving tape. This allows it to storeseveral terabytes of data of non-volatile data on a small space, making it very cheap compared to higherlevel memories which can separately store less than a terabyte. This affordability does come at the costof speed, since the turning of the medium takes time. Historically, magnetic storage has always beenused for low-frequency but high-capacity storage in e.g. servers or data banks [3].

By cleverly applying these memory types, it is possible to reduce the gap between memory and processor per-formance. However, the demand for larger and faster memories grows, and CMOS technology is hitting wallsthat prevent it from becoming faster and smaller [61]. Fortunately, new technologies are still emerging thatcan fit between levels of the memory hierarchy and close the gaps [72]. One of these promising technologiesis the use of resistive switching to build non-volatile memory [101].

2.2. Emerging memory technologies and RRAM designIn the last decade, efforts to further close the memory gap have brought forth several new technologies thatcan fill the level between DRAM and Flash memory, or even replace it [101]. One of these technologies isResistive RAM (RRAM) and is the main focus of this work. In this section, a brief overview will be given onemerging non-volatile memory technologies, followed by a more elaborate description of RRAM operation.

8 2. Background

Emerging non-volatile memory technologiesThe three most prominent types of emerging memories are Spin-Transfer-Torque Magnetic RAM (STT-MRAM)[52], Phase-Change RAM (PCRAM) [98] and Resistive RAM (RRAM or RRAM)[97]. All of them use resistiveswitching to switch between high and low resistance states. A consequence of this is hysteresis, which meansthat once a boundary voltage is crossed, the device quickly switches to another resistance, recording the his-tory of the applied voltage. The following list shortly describes the principles behind the three technologies.

Figure 2.3: The two common layouts of STT-MRAM: in-plane (left) and perpendicular (right). The reference layer consists of two layersof opposite polarity to increase stability [52].

• STT-MRAM [52], as displayed in Figure 2.3, uses two magnetic layers separated by a very thin (10Å)oxide layer called a magnetic tunnel junction (MTJ). One of the layers, called the free layer, is usedto store data by changing its magnetic state. The other layer is called the reference layer and retainsa stable magnetic state. The reference layer often consists of two layers of opposite magnetic fielddirection to increase the stability of the device. The relative orientation of the magnetic fields of thefree layer and the reference layer influences the resistance of the device, called the tunneling magneto-resistance (TMR) [106]. A state is written to the device using the spin-transfer torque (STT) effect [13].The direction of the fields can be either parallel to the plane in which the device exists (in-plane) orperpendicular to it (perpendicular).

Figure 2.4: The two common layouts of PCRAM: mushroom-shaped (left) and column-shaped (right)

• PCRAM [98], displayed in Figure 2.4, uses the change in phase of a semiconductor material to switchbetween resistances and store data in that manner. The phase-changing material is put in between twoelectrodes, one of which has a narrow connection to the material called the heater. By applying a largecurrent pulse, the semiconductor material melts, changing from a crystalline phase into an amorphousphase and increasing the device resistance significantly. Applying a longer pulse of medium magnitudeanneals the amorphous material back into a crystalline state, reducing its resistance again. The ma-terial can be wider than the heater, causing a mushroom-shaped amorphous region when melted, orconfined to a column.


• RRAM [97] uses a controlled dielectric breakdown to create a conductive filament to switch betweenresistance states. The device is a layer of oxide between two electrodes. The filament can be rupturedand regrown to switch between resistance states. Since this work concerns RRAM, the device will beexplained more extensively in the next section.

Principles of RRAMResistive RAM (abbreviated as either RRAM or ReRAM) is manufactured by the formation of a filament inan insulator between two electrodes, called a metal-insulator-metal stack (MIM stack) [97]. The length andwidth of the filament determine the resistance of the device. The filament can then be ruptured and regrownagain to change and switch the resistance of the device. This phenomenom has already been known andresearched 60 years ago [45] but the resistive switching was not robust enough for use in memory systems[97]. In the late 1990s to 2000s, resistive switching memories regained interest [6] followed quickly by a worldpremiere RRAM NVM implementation by Samsung Electronics [4]. HP Labs [86] then discovered that theRRAM device exhibits memristive behavior and invented the world’s first nano-scale memristor, opening up arange of new possibilities such as computation-in-memory [60, 104] and arrays for reconfigurable computing[23, 25, 107].

There are two technologies available to create a conductive filament: either by the dielectric breakdown ofa metal oxide and the drift of oxygen ions (OxRAM), or by the fast diffusion of metal ions into a solid electrolyteto create a conductive bridge (CBRAM) [94, 97, 101]. Figure 2.5 shows the two available RRAM technologies.OxRAM and CBRAM have similar characteristics, but the most notable difference is that OxRAM has a smallerdifference between resistance states, but a larger endurance [97]. This work will only consider OxRAM anddescribe the forming, reading and writing procedures of this technology.

Figure 2.5: The two available RRAM technologies: oxygen vacancy powered RRAM (OxRAM, left) and conductive bridge RRAM (CBRAM,right)

The oxide layer of an OxRAM device is a metal oxide, which means that it consists of metal atoms andoxygen atoms, arranged in a certain molecular structure [97]. This structure is not perfect [10] and at somepoints, oxygen atoms are missing, causing positively charged oxygen vacancies. These vacancies can captureand emit electrons and thus conduct current. An externally applied electric field can push more oxygen ionsaway from their positions in the lattice, further decreasing the resistance of the oxide. This process is calleddielectric breakdown, as it breaks down the insulating properties of the oxide, or dielectric. For the purposeof OxRAM, however, this is the mechanism that enables resistive switching.

The full operation process of an OxRAM device, consisting of initial forming and subsequent switching,can be described in six phases: Unformed device, forming the filament, low resistance state (LRS), the RESETprocess, high resistance state (HRS) and the SET process. The phases of operation are displayed in Figure 2.6.The letters A to F on the I-V curve correspond to the device states on the right and will be explained next.

A Unformed device: Directly after the MIM stack is manufactured, depending on the process [7, 51, 99],the unformed or “virgin” device already has a few oxygen vacancies present in the metal oxide. Thesenatural vacancies exist on the boundaries between grains in the polycrystalline structure of the metaloxide [10] which already conduct current and cause leakage paths [9, 59, 91]. This leakage current is,however, still very small.

10 2. Background

Figure 2.6: The full resistive switching process of an RRAM device of the OxRAM type, using bipolar operation. The letters A to F on theI-V curve on the left correspond to the respectively labeled figures of the device state on the right. [97]

B Soft breakdown: To start the forming process, characterized by the soft dielectric breakdown of theoxide, a relatively large voltage is applied to the MIM stack. This creates a large electric field insidethe oxide, pushing negatively charged oxygen ions out of their positions in the oxide lattice [97]. Thepositively charged holes, or vacancies, left by the ions, can conduct current [59, 91]. The oxygen ionspushed out of their positions drift through interstitial positions to a capping layer (the light gray layerstoring oxygen ions, underneath the top electrode in Figure 2.6), which acts as an oxygen reservoir.

The current flowing through the newly formed vacancies heats up the filament, which in turn speeds upthe generation of new vacancies, which again conduct more current [76]. This positive feedback loopcauses a sudden jump in current, caused by the abrupt forming of the filament in the oxide. As soon asthe current crosses a compliance value, the forming process is completed and . The abruptness of thedielectric breakdown is a source of variability for the RRAM device [19, 29, 37] and forming strategiesthat minimize this variation are often a subject of research [38, 41, 65, 66, 77, 81, 103]

C LRS: After forming, the device is ready for use and exists in low resistance state. This state is interpretedas a logical 1, and can be read by applying a relatively small voltage and measuring the resulting current.

D RESET process: To reset the device, a relatively large reverse bias voltage is applied, pushing the oxygenions from the reservoir back into the oxide. The ions drift towards the vacancies they left behind duringthe forming of the filament and reoxidize them, creating a gap in the filament. Only part of the filamentneeds to be oxidized to set the device to a logical 0 [97].

E HRS: The device is now in its high resistance state, interpreted as a logical 0. The filament is not entirelygone, but now a gap exists between the filament and one electrode. This gap increases the resistance ofthe device by several orders of magnitude [97]. Reading the 0 with the device in HRS works similar to inLRS, by applying a small voltage and measuring the resulting current.


F SET process Now, a relatively large forward bias voltage is applied, larger than a read voltage, butsmaller than a forming voltage. The voltage needs to be large enough to regrow the filament, but not aslarge as the forming voltage because part of the filament already exists. The heat of the (HRS) filamentcombined with the applied electric field once again push out the oxygen ions towards the reservoir inthe anode, similar to the final steps of dielectric breakdown in phase B. After a SET compliance currentis reached, the filament is rebuilt and the device returns to the LRS state in phase C.

An RRAM device can have two modes of operation, depending on the type of materials used to manufac-ture the device. One is unipolar operation, where setting and resetting the device is only determined by themagnitude of the applied voltage. The other is bipolar operation, where setting and resetting depends bothon the magnitude and the polarity of the applied voltage. The set and reset operations have reverse polaritiesin this case [97]. The electrical I-V characteristics of both modes of operation are displayed in Figure 2.7. Thiswork will only consider devices that use bipolar operation.

Figure 2.7: The I-V characteristics of two types of RRAM operation: unipolar (left) where the SET and RESET voltages are independent ofpolarity, and bipolar (right) where the SET and RESET voltages have opposite polarities [97]

A unique capability of RRAM is the fact that it acts like a memristor [24, 86] and can thus be used formore applications than just non-volatile memory [95], such as computation-in-memory [60, 104], arrays forreconfigurable computing [23, 25, 107], hardware security [17] and neuromorphic computing [49].

Implementation of emerging NVM in a memory arrayThe advantages of the emerging NVM technologies discussed and compared in Section 2.2 make them agood candidate for high density non-volatile memory arrays with the potential to contend with existing Flashtechnology. Whereas previously the focus was mostly on the devices by itself, this section will provide anoverview of the most common structures used to implement the devices into a larger context to create amemory array. These methods are: one-resistor (1R) or cross-point arrays, one-diode-one-resistor (1D1R)and one-transistor-one-resistor (1T1R) arrays [101, 102]. It should be noted that the “resistor” in contextrefers to the resistance switching element, i.e. the NVM device. Figure 2.8 shows the schematics of 1R and1T1R arrays.

• 1R arrays or cross-point arrays (Figure 2.8a) use just the NVM device, sandwiched between a crossbarstructure of vias. The word lines (WL), used to select a row from the array, run perpendicular to the bitlines (BL), used to select a bit or column from a row. An advantage of this structure is that it is simpleand small, since a memory cell consists of just the NVM device. The disadvantage is that selecting asingle device is not trivial [101] and other devices on the same word line are influenced by a read orwrite signal, causing unintended writes [31, 102]. Furthermore, without a selector blocking the path toa memory cell, current can run from the bit line, through other cells, back to the word line. This issueis called sneak path and can also cause incorrect reads, as multiple cells contribute to the measuredresistance [50].

• 1D1R arrays have a diode connected in series with the NVM device [63]. This solves some of the prob-lems of 1R arrays: current can no longer flow back up another device, eliminating the problem of sneakpath. However, the structure is larger and allows for less memory array density. Furthermore, since thecurrent through the device can only flow in one direction, the device must be compatible with unipolaroperation.

12 2. Background

(a) 1R design (b) 1T1R design

Figure 2.8: The two most common array design for NVM devices: one-resistor (1R) and one-transistor-one-resistor (1T1R) [101, 102].The figures show how the memory cells are connected to the bit lines (BL), word lines (WL) and, if applicable, the select lines (SL).

• 1T1R arrays, finally, solve most of the problems of 1R arrays by adding a transistor in series with theNVM device that acts as a selector (Figure 2.8b). The selector can be activated by sending a signal tothe word line (WL), which opens the NMOS transistor and allows a current to run through the NVMdevice. 1T1R arrays are more robust, but the requirement of an additional transistor make them moreexpensive: the structure includes a CMOS transistor which requires it to be connected to the substrateon the bottom of the chip, reducing the density of the memory array and making implementation as3D arrays more difficult.

2.3. RRAM manufacturing processThe manufacturing of RRAM arrays can be summarized in three steps [31]: front end of line (FEOL), back endof line (BEOL) and the forming process. The FEOL is the first part of the manufacturing process and buildsthe MOSFET transistors. The BEOL is the second step and adds all the metal layers of vias that connect thetransistors in the FEOL. The RRAM device itself exists in the BEOL, as a metal oxide between two metal layers.Finally, the forming process concerns step B in Figure 2.6, where the conductive filament is formed.

FEOLThe front end of line manufacturing process builds the substrate that holds the transistors of a chip. Thispart of the process has been important in the production of chips for decades. As feature sizes got smallerand smaller, following Moore’s Law [83] managing process variations has become more and more of a barrierto overcome [55, 56]. These variations include historical variations, that have been an issue and will stillcause issues, and emerging variations, that were historically insignificant but now play a role. Examples ofhistorical variations are patterning proximity effects, line-edge roughness, polish variations and variationsin the gate dielectric [56]. Emerging variations are the results of downscaling the transistors down to a fewtens of nanometers. Before downscaling, these variations were insignificant because they were averaged overa larger feature size, but in the last decade, transistors have become so small that new issues emerged [56].Examples are random dopant fluctuation, variation associated with implants and anneals and granularityvariations [56]. Even though these defects have little effect on RRAM devices - since these are manufacturedin the BEOL - it is worth noting that the transistors that coexist with RRAM are also imperfect.

BEOLThe back end of line manufacturing process builds several metal layers in a pattern that connects the tran-sistors below. The RRAM devices are also fabricated in these layers. The process is still equal to the originalprocess, so many defects are already known [30]. Misalignment of the layers or small particles can cause poorconnections that increase the resistance of a wire. Furthermore, line-edge roughness causes irregular shapesto form which affect the wire resistance and capacitance [87].

The first step in the fabrication of the actual RRAM devices is the patterning of the bottom electrode (BE).The process leaves a rough surface that causes variability between devices. This roughness can be mitigated

2.3. RRAM manufacturing process 13

by a chemical-mechanical polishing step [21], but this can still leave polish variations. The second step is thedeposition of the oxide layer. The thickness of the oxide and the amount of vacancies in it are also a sourceof variability and need to be carefully controlled. The molecular structure of the oxide also plays a role: itcan be either poly-crystalline (multiple, separate, unaligned grains of crystalline structures) or amorphous(no or very little crystalline structure) [39]. The disadvantage of poly-crystalline structures is that the grainboundaries, which are randomly distributed over the oxide, influence the variability of the devices [9]. Theadvantage is however that the difference between HRS and LRS resistances is much larger than in an amor-phous oxide [40] which makes it easier to distinguish between the logical states 1 and 0. After that, a cappinglayer is deposited on the oxide, which acts as an oxygen reservoir, capturing the drifting oxygen ions duringswitching operation. This improves the quality of switching [29]. The capping layer can be engineered to bea more efficient reservoir, or just be a part of the top electrode. In the last step of the fabrication process, thetop electrode (TE) is deposited, a process which is similar to the deposition of the BE, and experiences thesame types of variations and defects.

FormingThe final step of the manufacturing process happens after the device is already completely fabricated and isalso the step that is prone to the most variability. Even though the MIM stack is already built, the forming stepis still considered part of the manufacturing process, because it determines the characteristics of the RRAMdevices [30]. Ideally, a conductive filament is thin enough to be easily ruptured, but not too thin to still beable to conduct [18, 19, 66, 73, 74]. To achieve this, several special forming strategies have been invented,often called “soft forming” [41, 66, 81]. The three most common forming strategies are:

• Single pulse: (Figure 2.9) A single pulse of the forming voltage V f or m is applied for t f or m seconds. Afinite rise and fall time is necessary to avoid current overshoot effects [36]. This effect exists with otherpulse-based forming strategies as well, requiring the pulses to have a trapezoidal shape.

Figure 2.9: Single pulse forming: voltage (y-axis) vs. time (x-axis)

• Incremental form (IF) or Ramp forming: (Figure 2.10) Multiple pulses of length tpul se are appliedsequentially, increasing in voltage until a limit V f or m .

• Incremental form and verify (IFV): (Figure 2.11) Similar to IF, multiple pulses are applied, but afterevery pulse, the resistance of the device is measured using a small voltage. This strategy can be used tovery accurately control the filament shape narrow the spread of resistances in RRAM devices.

In comparison, IFV forming results in the best quality filament, while single pulse forming is the leastcomplex to implement. One should take into account that for commercialization, complexity can be a show-stopper: the IFV process requires a read in between pulses, which can only consider one device at a time andtherefore must be implemented serially. A full IFV process for a single device takes around 3.6ms [41]. Form-ing a memory array of 1GB thus takes up more than 42 days. So even though IFV produces a more controllablefilament, the commercially viable choice is a one-way system that can form multiple devices in parallel.

After the filament has been formed, the RRAM device is ready for use.

14 2. Background

Figure 2.10: Incremental forming: voltage (y-axis) vs. time (x-axis)

Figure 2.11: Incremental Form and Verify: voltage (y-axis) vs. time (x-axis). As soon as the measured current crosses a compliance value(Icomp ) the forming process is stopped.

2.4. DefectsNow that the manufacturing process of an RRAM device is clear, a few categories of defects can be extractedto be translated into a physical model. Based on the process variations that cause manufacturing defects, themodel will be adapted to investigate its effects.

It should be noted that much literature uses the term “defects” to describe lattice defects [8, 9,12, 22, 91]. To prevent any confusion, this work will refer to lattice defects only as vacancies anduse the term “defect” for a process variation induced deviation from the intended operation ofthe device. For clarity: this implies that a “defect-free” MIM stack still contains oxygen vacancies,because even though these vacancies are considered defective in its use as an insulator, they are acrucial part in its use as an RRAM device.

The process variations in the manufacturing process of the RRAM device that cause variations betweendevices and subsequently defects can be summarized into four categories [16, 81]: vacancy density variation,oxide thickness variation, electrode roughness and impurities. An overview of the four categories is givenin Table 2.1. The table lists the variation that causes defects (defect source), its realization in a model andthe (hypothetical) consequences of said variation. The following paragraphs elaborate on the contents of theoverview table.

Vacancy density variationLocal variation in the density of vacancies is already known in transistors as random dopant fluctuation (RDF)[56]. Vacancies tend to exist on the boundaries of crystal grains (GBs) which are spread randomly through thesurface of the oxide [9, 67]. Multiple GBs in a single MIM stack imply a larger vacancy density and could causemultiple or thicker filaments to grow, decreasing LRS resistance. Similarly, a smaller vacancy density couldcause an underformed or even missing filament. Implementation of vacancy density variation in a model isachieved by modifying the density of vacancies (NV +) in the initial lattice.

2.4. Defects 15

Defect source Realization and units in modeling Hypothetical consequences

Vacancy density NV + [cm−3] Overforming (density too high) under-forming or no forming (density too low)

Oxide thickness tox [nm] Overforming (oxide too thin) or under-forming (oxide too thick)

Electrode roughness σel [nm] Overforming (electrodes locally too closetogether) or underforming (GB locatedon locally far-spaced electrodes)

Impurities material, size, shape, position Overforming, underforming or no form-ing (depending on impurity characteris-tics)

Table 2.1: An overview of the four different defect sources resulting from process variation in the manufacturing process of RRAM devices[16, 56].

Oxide thickness variationProcess variations [55, 56] during the manufacturing of the MIM stack can also cause the metal oxide to bethicker or thinner than intended. A thicker oxide conducts less current and has a smaller internal electricfield, causing an underformed filaments. Likewise, a thinner oxide conducts more current and holds a largerelectric field, causing overforming [1, 16]. Oxide thickness variation can be implemented by varying the thick-ness of the oxide (tox ) in the initial lattice.

Electrode roughnessThe surface of electrodes are also not ideally flat, but show local fluctuations in thickness due to roughness[16, 56, 81]. This variation was already researched before RRAM [48, 54, 87] and can be modeled as a sinu-soidal deviation [54] or a stochastic deviation [16, 48, 81]. Electrode roughness could cause the oxide to locallybe quite thin, causing overforming of the filament. Both sinusoidal and stochastic implementations of elec-trode roughness can be characterized by a standard deviation (σel ) from the planar oxide-electrode surfacesin the initial lattice.

ImpuritiesFinally, any random shape of any material could be contaminating any part of the stack [16]. The severityof the defect is determined by the properties of the impurity. The effects of impurities on the formation of afilament and subsequent RRAM operation are as of yet largely unresearched. Impurities can be implementedin a model by changing the characteristics of regions of the initial lattice. The shape and properties of saidregions depend on the material, size, shape and position of the impurity.

16 2. Background

Figure 2.12: A schematic overview of the four different categories of defect sources: vacancy density, oxide thickness, electrode roughnessand impurities.

3State of the art

This chapter provides an overview of the many existing models that describe various aspects of RRAM oper-ation to research their effects. The focus of this chapter lies on the modeling of defects, but the many studiesthat look at the variability of defect-free devices are also taken into account. First, a few existing studies onRRAM defect modeling are listed and explained. Since these models have shortcomings in terms of phys-ical accuracy, in this work, a physical defect-free model is adapted to include defects. For that purpose, aclassification of defect-free RRAM models is given, after which a few models are listed that fall under saidclassification. Finally, the defect-free models are compared and one is chosen from the listed models, basedon a number of requirements that make it the best fit to include defects.

3.1. RRAM defect modelingIn this section, three existing studies on RRAM defect modeling are listed and explained. Even though theinherent variability of RRAM devices has been a subject of research for over a decade, the effects of defectshave only been a main focus in the past few years. The three studies that do take RRAM defects into accountare Chaudhuri and Chakrabarty ([16]), Fieback et al. ([32]) and Abbaspour et al. ([1]).

Chaudhuri and Chakrabarty, 2018Chaudhuri and Chakrabarty ([16]) present a physics-based classification and analysis of memristor fault ori-gins. Its motivation is the disconnection between existing fault models and the underlying physics of RRAMdevices. A one-dimensional electrical model [108] is used, only modeling the position of the boundary be-tween doped (filament) and undoped (without filament) oxide. The fault-free operation of the model isdemonstrated and verified. Then, 5 process variation-induced fault origins are identified and their varia-tion range is divided in two categories: catastrophic (causing stuck-at faults) and benign (no effect). Plots ofthe process variations and their categorizations are displayed in Figure 3.1.

Using the electrical model, the variations are observed and categorized. After that, the effect of breaks,voids and punch-throughs on the resistance of the electrodes is investigated. While this does give a goodindication of the effects of these defects, this work draws its conclusions from adapting parameters of anelectrical model. This electrical model is unable to model the shape of the filament which could also beinfluenced by defects. Therefore, a lower level model is needed to verify the defective behavior of this compactmodel.

Fieback et al., 2018Fieback et al. ([32]) Proposes a new form of defect modeling and subsequent test development, motivatedby the shortcomings of cell-aware testing, a traditional method of simulating defects by addition of linearcircuit components next to the device under test. Device-aware testing (DAT) considers instead the physicalcharacteristics of the defective device itself, rather than describing its consequences with electrical compo-nents (e.g., linear resistors) that surround a defect-free device. The DAT-process consists of three steps: defectmodeling, fault modeling and test development, which help describe defects that are unique to RRAM (andSTT-MRAM, which is also considered in this work) as detectable faults, to prevent test escapes. The processof DAT is displayed in Figure 3.2.

17

18 3. State of the art

Figure 3.1: The results of Chaudhuri and Chakrabarty ([16]), a physics-based classification and analysis of RRAM fault origins. The graphsa) to f) show the effects of the labelled process variations on the minimum write-1 duration. A green shading indicates benign variation,a red shading indicates catastrophic variation, and an orange shading indicates the transition region from benign to catastrophic.

Figure 3.2: The main principle behind Device Aware Testing (DAT) as a flowchart, as proposed in Fieback et al. ([32]). The three steps ofdefect modeling, electrical modeling and then fitting are shown to lead to an optimized defective device model.

The model that is used for RRAM defects directly translates technology parameters (e.g. oxide thickness,conductive filament dimensions) to electrical parameters (e.g. SET/RESET voltage threshold, HRS/LRS resis-tance) through a series of equations, fitted to experimental data. The technology parameters are modifiedto reflect defects, after which implementation context is added by putting the device in a Cadence Spectrenetlist. With the new defective model, tests are developed that can sensitize the unique RRAM faults.

DAT is however more based on the electrical properties of defects, and does not take into account thelower level physical properties such as the shape of the filament.

Abbaspour, 2018Abbaspour et al. ([1]) Presents a physical model to investigate the full resistive switching process of OxRAM.The model is three-dimensional and uses a kinetic Monte Carlo method to simulate the forming and disso-lution of a vacancy-rich filament. This model is described in further detail in Section 3.3.

The main focus of this work is not defect analysis, but in the application of the model, the effect of oxidethickness on the forming operation is investigated, which is a possible source of defects. The resulting effect

3.2. Classification of defect-free RRAM models 19

Figure 3.3: The effect of different oxide thicknesses on the forming process, resulting from the kMC model by Abbaspour et al. ([1]). Aforming voltage threshold was extracted from the I-V curves and plotted on the bottom right.

on the forming I-V curve is displayed in Figure 3.3. Although, while this model does use a lower level physicalimplementation than the other two models described above, it lacks a specific focus on defective behaviour.

Looking at the current state of the art, it seems that in-depth research of the physical effects of defectsin RRAM is still missing. Chaudhuri and Chakrabarty and Fieback et al. do focus on the effect of these de-fects, but base their defect simulations on modifications of parameters in fitted equations. Meanwhile, manymodels like Abbaspour et al. exist [1, 43, 57, 69, 76, 82] that describe the formation of a filament using phys-ical processes, but leave defects as a side focus. A connection between low-level physical descriptions ofthe RRAM resistive switching process and high-level defect analysis could provide a valuable insight into thephysical effect of defects, and rigor to the fittings and assumptions made in higher level models.

Creating such a model from scratch is not feasible - and unnecessary indeed, given the large number of(defect-free) models that are already available in the state of the art. Rather, an existing model should beelaborated to include defects that can be tuned to research their effects. Such a model must fit a number ofrequirements:

1. The full resistive switching process must be modeled. Forming, SET and RESET all contribute to thecreation of defects. If the effect of physical defects on electrical characteristics is to be shown, then themodel must be able to simulate electrical operation.

2. It must be possible modify to the shape of the device in detail. In order to test the effects of defectssuch as electrode roughness or impurities, the shape and material of various parts of the model mustbe adaptable.

3. The shape and position of the filament cannot be assumed. Defects can alter the shape and position ofthe conductive filament, which might contribute to the creation of defects. Therefore, a pre-definitionof a filament shape - often as a trapezoid in the middle of the oxide - cannot be assumed.

4. The model must be reproducable. Given the information provided in the work describing the model,it must be reproducable. Unreferenced dedicated software or missing information prevents the con-struction of a working and verified model.

With these requirements in mind, a defect-free model can now be chosen from those available in the stateof the art.to be modified to include defects. To provide orientation, a classification of models is necessary.

3.2. Classification of defect-free RRAM modelsGiven the large number of defect-free RRAM models in the state of the art, it is helpful to classify them de-pending on their scale and intended use. Lanza et al. have proposed a classification into three groups basedon the level of abstraction of the physical processes involved in the operation of RRAM. These three classes areMicroscopic, Macroscopic and Compact. A symbolic representation of the abstraction level and simulationscope of the classes is displayed in Figure 3.4.

The description of these classes are:


Figure 3.4: The classification of state-of-the-art resistive device models [57]: Microscopic, Macroscopic and Compact. Models on the leftof the scale are more complex and small-scale, but physically accurate. Models on the right of the scale are less accurate on a physicallevel, but relatively fast and simple.

• Microscopic models: this type of model, also called an atomistic model, looks at atomic-level physicsand interactions and attempts to describe them from scratch, a method called ab initio modeling. Theyconsist of three main components:

1. The relevant phenomena from atomic physics and material science that have proven to enableresistive switching: such as crystal lattice structures, electron localization functions [27] and po-laronic interactions [109];

2. An electron and ion charge transport model;

3. A kinetic Monte Carlo engine to simulate the evolution of said phenomena over time.

Other than models of a higher level of abstraction, microscopic models try to use physical characteris-tics obtained from measurements to describe phenomena in as much detail as possible. Thanks to thedeep level description of the relevant processes, the model can be calibrated to experimental data tosimulate the complex physics of resistive switching and explain them with processes and models thatare already familiar. The results obtained from this type of model can then be used to calibrate modelsof a higher level.

Pros: The most accurate model on an atomic scale. Describes RRAM physical processes using theoret-ical physics and uses it to attempt to explain measured characteristics.

Cons: The detailed description of the physical processes on a small scale (just a few hundred atoms[27, 28, 109]) is too complex to simulate the context of RRAM operation. Increasing the scale to cover afull device is computationally intensive and impractical for its purpose.

• Macroscopic models: To simulate the electrical characteristics of resistive switching, the atomisticphysical processes must be approximated by less computationally intensive alternatives, increasingthe scale of the model. This allows the model to simulate the full operation process of a single RRAMdevice, based on results obtained from both experimental data and microscopic models. This combi-nation of precise knowledge of physical processes in an RRAM device and calibration with electricalmeasurements from experiments makes it a powerful tool to simulate the full resistive switching pro-cess.

Lanza et al. calls this group the kinetic Monte Carlo/Finite Element Method (kMC/FEM) group, becauseboth methods play an important role in this class of models. The finite element method is used in thecalculation of potential profiles and temperature through Poisson and Fourier equations, and kineticMonte Carlo is used to determine the evolution of the FEM state over time, similar to microscopicmodels but on a larger scale.

Pros: A combination of knowledge from the results of microscopic models with calibration to electricalmeasurements to fit the device’s characteristics makes a macroscopic model a good tool to simulate the

3.3. Macroscopic models 21

full RRAM switching process in detail. It provides information about electrical characteristics such asforming, set and reset voltage, and also shows the physical shape of a filament.

Cons: Approximations of physical specifics from microscopic models might not be completely accuratecompared to realistic situations. On the other hand, the finite element method requires a large list ofpoints to model the molecular lattice, making simulation of large RRAM arrays impractical.

• Compact models: Also called semi-empirical models, this class of models mostly uses the electricalcharacteristics of an RRAM device. They run in SPICE-like simulation environments and rely on thesimplification of the conductive filament to, e.g., an Ohmic conductor, a quantum-point contact (QPC),or a space-charge region. Using these simplifications, the models are fitted to experimental data andmacroscopic models. Some models even neglect the physical operation of an RRAM device and focusonly on the reproduction of experimental data [84].

Pros: The simplification of the physics of an RRAM device to only its electrical characteristics allowscompact models to simulate many devices in parallel. Furthermore, their compatibility with SPICEallows them to be included in a complete circuit that also accounts for, e.g., interconnect and senseamplifiers.

Cons: The physical details of the resistive switching process are lost in the simplification to a compactmodel. It is therefore not fit for studying specifics such as, e.g., the forming of a filament.

In conclusion, the three classes of RRAM models have different levels of abstraction which makes themuseful for different purposes. Microscopic models are best for explaining physical processes involved in re-sistive switching with theoretical physics. Macroscopic models are best for bridging the gap between physicsand electrical characteristics. Finally, compact models are best for looking at RRAM devices in context.

Now that a model classification is established, the requirements of the model that were listed in Sec-tion 3.1 can be used to choose an appropriate model class for this work. Table 3.1 compares the model classeswith the strengths and weaknesses if each model class in mind.

From Table 3.1 it can be concluded that to find a defect-free model that fits the requirements from Sec-tion 3.1, a macroscopic model is needed. Microscopic and compact models can instead be used as referencesto retrieve information and verification required to build the macroscopic model.

3.3. Macroscopic modelsThis section will list the existing state of the art RRAM models, specifically those that are considered macro-scopic models, and briefly address their main focus, method, conclusion and usability for this work. Theusability is based on an argumentation given the requirements listed in the previous section.

Guan et al., 2012 [43]

Main focus To study the variation in switching parameters in metal-oxide-based RRAM.Method A physics-based model capable of simulating a large number (>1000) of switching cycles.

The model is 2D to improve performance, but does take into account the physics in thethird dimension. Trap Assisted Tunneling is the dominant conduction method in HRS. InLRS, the shortest path through a filament is found and its resistance determined. Genera-tion and recombination of vacancies is determined by probabilities.

Conclusion A correlation is shown between the I-V curves observed at the device terminals and thestochastic vacancy dynamics in the oxide, and it is verified by measurements. Also showsthe variation in resistance of HRS and LRS over multiple cycles.

Usability This model appears to be simple to reproduce and is not computationally intensive. Itdoes, however, make some assumptions, such as using only two dimensions, which mightobfuscate interesting characteristics of filament forming in a defective device.


Table 3.1: A comparison of the three model classes presented in [57], using the requirements listed in Section 3.1. Red shading impliesthat the class does not meet the requirement. No shading implies that the class partially meets the requirement. Green shading impliesthat the class always meets the requirement.

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Microscopic Only parts of the RSprocess are modelledin a very small scope.Electrical character-istics such as appliedvoltage over time arenot considered.

The shape of the modelspace can be detaileddown to the shape of theatoms involved.

The filament shape - ifeven considered - is nota product of a stochasticprocess but pre-defined[27, 109].

Microscopic models areoften reproducible butrequire an in-depth un-derstanding of theoret-ical physics to imple-ment correctly.

Macroscopic Simplifications of phys-ical processes, linkedto RS by microscopicmodels, allow the sim-ulation of one or morefull RS cycles.

The FEM implies thatthe model space can bemodified within the de-fined resolution of thefinite elements. Withinthis resolution, any de-tailing is possible.

Due to the use of kMCmethods, the evolutionof the shape of the fila-ment is defined only bystochastic events overtime.

Whereas ease of re-producibility variesbetween availiablemacroscopic mod-els, they are mostlybased on a few clearlydocumented processes.

Compact The simplicity of com-pact models allowsthem to simulate manyRS cycles for multipledevices.

Large-scale shapedetails such as oxidethickness can usuallybe modified, but theeffect of small-scalevariations such aselectrode roughnessor impurities must bebased on assumptions.

The shape of thefilament is always pre-defined and can onlyvary in dimensions,such as thickness or gapwidth.

Compact models arebased on a few explicitequations, but also relyheavily on empiricaldata.

Raghavan et al., 2014 [81]

Main focus To investigate the impact of grain boundaries, process-induced vacancy distribution,metal-dielectric interface roughness and multi-layer high-κ films on the variability offorming.

Method A Monte Carlo model with a 2D cell-based percolation model. Vacancies are generatedbased on a generation rate λG . The model only considers the generation of vacancies andthe influence of process variations on it, but does not consider the current through it.

Conclusion A link was found between process variations, such as electrode roughness and vacancydistribution, and their effect on forming time. It is shown that the presence of grain bound-aries reduce the power required for forming, but increase the forming variability.

Usability The study of process variations is a useful aspect of this work, but it does not include acharge transport model and the effect of current on vacancy generation. Furthermore,only generation rates are considered, so resetting the model is not possible. Therefore,while suitable for showing physical effects of defects, it is not useful for electrical analysis.

3.3. Macroscopic models 23

Sadi et al., 2015 [82]

Main focus To study the switching process of silicon-rich (SiOx ) RRAM devices.Method The model couples a kMC engine and an electron transport solver to the statistical 3D

TCAD simulator “GARAND”, which solves Poisson equations to retrieve electric field pro-files. The temperature is calculated by a more advanced in-house time-dependent heatdifferential equation solver.

Conclusion The results show that resistive switching is an intrinsic property of silicon oxide. Themodel is proposed to facilitate efficient RRAM designs in terms of performance.

Usability Even though this model shows promising results and is based on well documented equa-tions, the simulator “GARAND” is no longer available. The reproducibility of this model istherefore not ideal.

Menzel et al., 2015 [69]

Main focus To model resistive switching in electrochemical metallization cells (ECM), or ConductiveBridge RAM (CBRAM).

Method A kMC model that describes the atomic process of RS by four different rates: ion hopping,ion reduction, oxidation, and nucleation. The potential distribution is calculated as a 2Dfinite element model using COMSOL Multiphysics, accounting for the potential drop atthe electrode interfaces. The electron-transfer process for the tunneling current is calcu-lated using a linear equation. Also, the influence of mechanical stress on the shape of thefilament is studied.

Conclusion It concludes that the size and shape of the filament in a CBRAM device depends on theapplied voltage and mechanical stress. The work also mentions compact, 1D models bythe same authors, of which the results were now further verified by this kMC model.

Usability The model is documented clearly, with provided visualizations of the filament shape. Itdoes, however, consider CBRAM instead of OxRAM, whereas the focus of this work lies onOxRAM.

Padovani et al., 2015 [76]

Main focus To describe the forming and switching operations of HfOx RRAM at a microscopic level,and their dependence on electrical conditions and device characteristics.

Method As a charge transport model, a previously published model for Trap Assisted Tunnelingis used. The potential and temperature profiles are calculated with a Poisson’s equationsolver. The kMC engine considers three events: generation, drift and recombination.

Conclusion It demonstrates the dynamics of the forming process and the effect of external tempera-ture and current compliance on the switching process. In conclusion, it presents a modellinking microscopic material properties to electrical characteristics.

Usability Even though this model is named “microscopic” by the authors, it is in fact a macroscopicmodel as it links the physical processes of RRAM to its electrical characteristics. Beingclosely based on microscopic models makes it accurate, but possibly too complex to im-plement. However, this model has multiple, clear references amongst which are earlieriterations that it is based on [8, 10, 58, 90, 92].


Abbaspour et al., 2018 [1]Main focus To present an alternative model for investigating the switching cycle of OxRAM.

Method It uses a 3D kMC structure and only allows vacancies to form at the anode/oxide boundaryby exchanging oxygen to the anode, neglecting mobile interstitial oxygen ions. Instead,vacancies are considered the mobile species. The conduction method is Trap AssistedTunneling. The kMC engine uses three types of events: generation, recombination andvacancy drift. Using a voltage ramp, the switching process of forming, setting and resettingis simulated. It also investigates the effect of different oxide thicknesses.

Conclusion In conclusion, it reports the development of a model that could be used to simulate hun-dreds of switching cycles fitted to experimental data.

Usability The model is clearly documented, and appears simple enough to reproduce. It does, how-ever, simplify away interstitial oxygen ions that are the actual driving force behind recom-bination.

3.4. Comparison and defect-free model choiceFrom the six state of the art models, one model will be used as a reference for this work’s model to investigatethe effects of defects. The choice is based on the usability of the model.

The choice of reference model will be the model by Padovani et al. [76], because of its reproducibilityand physical accuracy. The other models are still good candidates, but either lack details on the model im-plementation, or are simplified in some way. Even though these simplifications are most often verified withexperiments, they do not take into account the effects that defects might cause. Therefore it is a good idea tochoose a model with as much connection with the underlying physical processes as possible.

4Model implementation

This chapter details the implementation of the RRAM model simulating the effect of defects on the formingof the filament and subsequent resistive switching. The model is implemented in and optimized for MATLAB,but the description presented in this chapter can fit any framework.

First, an overview is provided, describing the general structure of the reference model in [76] and the gen-eral structure of the model of this work, in Section 4.1. Then, the general modules of the model are explainedin Section 4.2.

After that, the chapter moves on to the implementation of actual theoretical RRAM physics. The firstmodule is the 3D Poisson equation solver, explained in Section 4.3. Then, the charge transport model, orTrap Assisted Tunneling (TAT) solver is described in Section 4.4. The last part of the reference model, thekinetic Monte Carlo (kMC) engine, is described in Section 4.5.

This concludes the implementation of the reference model. Section 4.6 then explains the additional mod-ule, that modifies the initial model lattice state to include manufacturing defects. The complete implemen-tation is available as MATLAB code at [47].

4.1. Overview of the model structureThis section gives an overview of both the reference model in [76] and this work’s MATLAB model.

Model structure: Padovani et al.Figure 4.1 shows the general structure of the model from Padovani et al.. It describes the steps of the loopingprocess the model takes from start to end. The two orange boxes indicate where results from microscopicmodels and measurements were used to provide information to this macroscopic model. The simulationsteps are displayed in the yellow boxes and are briefly elaborated below.

• Start: the simulation starts.

• Vacancies and ions distribution: before entering the calculation loop, the initial lattice is built by dis-tributing a number of vacancies and ions in a confined 3D space. Padovani’s model uses an oxide of10×10×10nm. The vacancies are spread uniformly and randomly throughout the oxide, according tovacancy densities measured in [78].

• 3D field map: a Poisson equation solver is used to calculate the 3D electric field map from the externallyapplied voltage and the local charge of vacancies and ions in the oxide.

• Current calculation: the current through the oxide is calculated, using a TAT solver (see also: Sec-tion 4.4) from another work [59, 91].

• IG >>> ICC , tSI M >>> tF (Check exit condition): stop the simulation if the current exceeds a compliance(ICC ) or the simulation time exceeds a maximum tF .

• 3D power dissipation and temperature: the TAT solver produces a power dissipation map, which isused to calculate the lattice temperature using the heat equation and a Poisson equation solver.

25

26 4. Model implementation

Figure 4.1: The model structure overview from Padovani et al. ([76]). The light yellow boxes represent steps of the model operation. Theblack arrows represent the flow of information in the operation of the model. The orange arrows and boxes represent places in the modelwhere results from microscopic simulations are used.

• V/O diffusion/generation/recombination rates: event rates are calculated for three types of events:the diffusion, generation and recombination of vacancies and ions.

• Event selection using Monte Carlo technique: a kMC engine (see also: Section 4.5) randomly selectsone of the events based on their rates and executes it.

• tSI M === tS I M +++∆t (Time step): based on the results of the kMC engine, a time step is added to thesimulation time.

This structure will be used as a reference to construct the base defect-free model in MATLAB for thiswork. Unfortunately, the original work and its references mention only the physical details, more practicalinformation such as used frameworks and optimizations is missing [10, 58, 76, 90, 92]. This must therefore befilled in.

Model structure: this workFigure 4.2 shows the general structure of the model implemented in this work, which will also be explained inthis chapter. The looping process is identical to the process described in the reference work [76] (Figure 4.1)but the structure in Figure 4.2 also mentions the more practical components of the model that connect thephysics modules and handle input, output and parameter sweeps.

The yellow boxes represent components that were reproduced directly from the reference model, whichwill henceforth be referred to as physics modules. The blue boxes represent the software that connects thephysics modules and manages inputs and outputs, the general modules. The general modules act as “glue” toconnect the physics modules into a full model. Finally, the red box is the defect injector, the main addition tothe model by this work.

The following list will briefly address each module of the structure and the interactions between the mod-ules.

• Model handler: (Section 4.2) accepts a set of instructions that describe the simulation the model isabout to perform, i.e. a variable and a set of values to create a sweep, or simply a single property. It theninitiates and controls the base model object to retrieve output variables such as I-V curves, grid layoutsand temperature profiles.

• Base model object: (Section 4.2) stores the lattice array and collects or distributes data to and from thelattice elements, to send to the physics modules according to the process specified in Figure 4.1. Every

4.1. Overview of the model structure 27

Figure 4.2: The model structure of this work.

step, it returns the current and the time step (and any other information the handler requests), andaccepts a new voltage value from the handler, reiterating the same loop.

• Lattice element object: (Section 4.2) represents a point in the lattice in the model space. Its type de-fines its function, such as a normal oxide, vacancy or electrode. Every lattice element also stores fieldssuch as its local potential or temperature. The properties of every element are updated according tocalculations from the physics modules.

• Variable plotter: plots a 3D block plot (e.g. Figure 5.2) of any property of all lattice elements.

• 3D Poisson solver: (Section 4.3) solves the Poisson equation in 3D to obtain temperature or potentialfields from respectively the power or charge distribution.

• TAT solver: (Section 4.4) calculates the current through the device based on the current state of themodel and returns it together with a dissipated power profile for all lattice elements.

• Kinetic Monte Carlo engine: (Section 4.5) picks a state changing event, i.e., diffusion, generation, orrecombination, based on the current state of the model and returns it with a corresponding time step.

• Defect injector: (Section 4.6) modifies the layout of the initial lattice grid according to (potentially)include a defect, described by parameters supplied by the handler, before the model starts running.

Together, these modules are able to model the operation of an RRAM device on a macroscopic scale thatcontains a manufacturing defect. In the next sections, the modules discussed in the previous list will bedescribed in detail.


4.2. General modulesThis section describes the general modules of the MATLAB model. They handle communication, input andoutput, and serve as the glue between the physics modules. The core general modules of the model structure,as displayed in Figure 4.2, are the model handler (source code function name: modelHandler), the basemodel object (model) and a point in the lattice grid (latticeElement). The variable plotter was mostlyused for debugging and imaging purposes and has no use in the model itself, so it will be omitted in thischapter. Examples of it in use can, however, be seen in Chapter 5 and the source code is available in thecomplete repository at [47].

Model handler (modelHandler)The modelHandler function serves as a wrapper for the base model object. Calling the function withoutarguments automatically sets up the model object with the default settings, simulating a perfectly manufac-tured RRAM device with default dimensions (see also: [47]). The function then handles three tasks:

• Determine the max time step. The applied voltage V (t ) is given as a MATLAB symbolic function (syms)dependent on the time t. The time step is determined by the kMC engine (see also: Section 4.5) and inturn depends again on the applied voltage, since a higher electric field in the oxide increases the eventrates. This cyclical dependency is problematic: when the applied voltage is very small, event rates -which depend on the externally applied field, ergo voltage - also become very small, which producesan event with a very large time step. This would make sense if the applied voltage did not change overtime; however, after some time the voltage might have increased, reducing the timestep. Therefore, thetime step of an event must be verified before executing it.

Figure 4.3: The process of picking a time step limit (dt_max). The symbolic function V (t ) is evaluated at the current state in time(blue) and, deducing from a max voltage step supplied by an argument (dV_max), a time step limit is calculated (orange). The kMCengine (Section 4.5) produces an event with a corresponding time step; if this step is within the limit, the event will be executed (green).Otherwise, it is discarded (red).

Figure 4.3 shows the process of deriving a time step limit. The handler arguments supply a maximumvoltage step, which defines the resolution of the simulation. MATLAB’s symbolic package is then usedto solve for the maximum time step to stay within the provided limits of voltage change. This time steplimit is provided to the kMC engine, which then decides whether or not to execute an event, dependingon the limit provided by the handler. If an event crosses the provided time limit, it is discarded.

• Execute a model step. With the derived time step limit, a step is executed using the doStep method ofthe base model object. This method completes one cycle of the flowchart displayed in Figure 4.1, usingthe components displayed in Figure 4.2. The step can result in a change of the lattice state, if the kMCevent rates are high enough, or otherwise continue without events. Eitherway, a time step is returned,which is added to the simulation time.

• Plot the model progress. When enabled (mostly used for testing and debugging purposes), the handlerfunction can display the state of the model and graph of several relevant variables, such as the currentover time or the temperature profile, using the plotGrid method of the base model object.

The simulation ends either when a compliance current is reached, or when the simulation time exceedsa maximum value. The handler function then automatically saves the full progress of the simulation and

4.2. General modules 29

returns the model object in its final state, which can then be reused, e.g. to simulate a full RS cycle afterforming.

Base model object (model)The base model object is the central controller that stores information on the model state and has severalmethods to handle the physics modules. It is defined in a class, using MATLAB Object Oriented Program-ming (OOP). The reason for this choice of implementation is that model now represents a physical object,i.e. the RRAM device. Its properties represent the state of the device, and its methods represent the actionsperformed on it, changing its state.

The fact that the base model object is a MATLAB OOP object also means that simulation time shouldbe managed elsewhere, to keep the implementation intuitive. The model object receives instructions froman external handler (modelHandler) which manages the time externally. Thus, a division between the deviceand its testing setup is made: model represents the device itself at a fixed moment in time, whilemodelHandlerrepresents the testing setup attached to its electrodes.

The most important methods of the model class are the following:

• model: the constructor method. It accepts a structure holding lattice properties, the applied voltageat the start of the simulation and the ambient temperature. It then builds the initial lattice, potentiallyincluding defects (see also Section 4.6) and returns a model object.

• doStep: executes a single step, traversing the flowchart in Figure 4.1 in the following order:

1. Update the potential and electric field profile according to the new model state and applied volt-age, using the Poisson equation solver (Section 4.3).

2. Calculate the current through the device and the dissipated power profile, using the TAT solver(Section 4.4).

3. Update the temperature profile according to the new power profile, using the Poisson equationsolver.

4. Calculate the time step and pick an event using the kMC engine (Section 4.5). If the event’s timestep is low enough, the event is executed.

5. Return the calculated current and the time step.

For debugging reasons, control over the model and other surveillance during model operation, themodel object can by itself only execute one step at a time. An external handler (modelHandler) is usedto perform the full simulation, while also keeping logs of the model’s state after every step.

• plotGrid: this method represents the variable plotter displayed in Figure 4.2. It accepts an optionalparameter, indicating a field of the objects in the lattice grid array (e.g. potential (phi), temperature (T),electric field (F), etc.) and plots the values of the parameter in a 3D block plot. A parameter is optional:without a parameter, the method plots the physical state of the model grid, showing the position ofe.g. vacancies, ions, and the electrodes (examples of plotGrid in use can be found in the results inChapter 5).

When the handler finishes the simulation, the model object is returned. It retains its final state, frozenin time, and can be used as the starting point for another simulation, or to investigate the properties of thesimulated device in detail, by using the methods and properties of the model class.

Lattice grid element (latticeElement)The core property of the model object is the grid. The grid consists of an array of latticeElement objectsthat each represent a physical point in the model lattice. They are arranged in a regularly spaced 3D grid andhave a number of fields and methods to store and modify their local state. The fields of a latticeElementare:

• x,y,z: the 3D position of the element in the model space.

• type: the type of the element, saved as a string which can be any one of the following:


– Regular oxide ('oxide')

– Oxide with an oxygen vacancy ('V+2')

– Oxide with an interstitial oxygen ion ('O−2')

– Top electrode ('upper')

– Bottom electrode ('lower')

• phi: the local electric potential.

• F_vec: the local electric field as a 3D vector.

• T: the local temperature.

• Q: the electrical charge of this element.

• P: the power dissipated in this element.

• E: the trap energy of this element (see also Section 4.4).

Note that, in the source code [47], a latticeElement has more fields, however, these fields are onlyrelevant to a specific part of the model (e.g. A_row for the Poisson solver) and will be mentioned at that partof the model to prevent confusion. The full list of fields is available in the source code [47].

The information stored in the fields of the latticeElement array represents the physical state of themodel and is therefore used to calculate the current and next event in the TAT solver and kMC engine. This isdisplayed in Figure 4.2 by the connections between the latticeElement block and the physical modules.

The next three sections will describe the implementation of the physical modules; the Poisson equationsolver, the TAT solver and the kMC engine.

4.3. Solving the discrete Poisson equation in 3DThe 3D Poisson equation solver, abbreviated as the Poisson solver, is crucial in determining the tempera-ture and electric potential distribution inside the model lattice grid. This section will explain how it works:first, the Poisson equation itself and its applications in the model are explained. Then, the Finite DifferenceMethod (FDM) is introduced and explained, which is crucial to applying the Poisson equation in a discretespace. Finally, the Conjugate Gradient (CG) method is explained, which is used to solve the discrete Poissonequation.

The Poisson equation and its applicationsThe Poisson equation is a differential equation, named after the French mathematician and physicist SiméonDenis Poisson. It appears in several forms in theoretical physics, but it always has the same general structure:

∇2φ= f (4.1)

In this general form of the Poisson equation, f is given, and φ is sought. ∇ (pronounced “nabla”) is thedivergence operator, which in 3D Cartesian coordinates takes the following form:

∇=[∂

∂x,∂

∂y,∂

∂z

]T

(4.2)

∇ is a vector operator, which is used on a vector field to retrieve a scalar field that represents the vol-ume density of the outward flux of the field. If multiplied with itself, however, as in the Poisson equation inEquation 4.1, it becomes a scalar operator:

∇2 =(∂2

∂x+ ∂2

∂y+ ∂2

∂z

)(4.3)

This operator is also known as the Laplace operator, written as ∆. Replacing ∇2 with its Cartesian formin Equation 4.1 and expanding it reveals that the Poisson equation is indeed of differential nature and thatsolving for φ is non-trivial:

4.3. Solving the discrete Poisson equation in 3D 31

∂2

∂xφ(x, y, z)+ ∂2

∂yφ(x, y, z)+ ∂2

∂zφ(x, y, z) = f (x, y, z) (4.4)

Like with any differential equation, the Poisson equation can only be solved if boundary conditions aregiven. This implies that within a defined boundary domain, φ(x, y, z) must have pre-determined boundaryvalues. In the model, this domain is usually defined by the shape of the electrodes, since the operation of themodel is controlled by the voltage applied on the electrodes.

In the RRAM model, the Poisson equation is used for two calculations: the electric potential field andthe temperature distribution. The electric potential φ(x, y, z) is related to the charge density ρ(x, y, z) anddielectric constant ε through Gauss’s Law in differential form:

∇2φ(x, y, z) =−ρ(x, y, z)

ε(4.5)

Similarly, the temperature T (x, y, z) is related to the power density P (x, y, z) and the thermal conductivitykth through the steady-state heat equation:

∇2T (x, y, z) = P (x, y, z)

kth(4.6)

Both Equation 4.5 and Equation 4.6 are simply Poisson equations and can therefore be solved in the exactsame manner, using the same module. Therefore, from now on, the right side of the equation is defined asthe given function f (x, y, z), and the sought function is defined as φ(x, y, z), producing the general Poissonequation as given in Equation 4.1.

The problem is, however, that the given function f (x, y, z) is determined byρ(x, y, z) or P (x, y, z), neither ofwhich can be described by a simple function. Furthermore, they are not described as an analytical function,but rather as an arbitrary collection of data points. Therefore a discrete solver of the Poisson equation mustbe implemented. To achieve this, the Finite Difference Method is used.

Finite Difference MethodThe Finite Difference Method (FDM) is a technique to find the derivative of equations that are defined not asan analytical function, but discretely. Discrete functions are a series of data points of a finite length. A discreteversion of φ is, for example, defined as N indexed points φi , for index i ∈ [1,2,3, . . . , N ]. Every discrete datapoint φi represents a sample from a continuous space. If these samples are equally spaced, they can beconnected to a continuous space by defining a step size. A 1D example, with a step size ∆x, is:

φi =φ(i∆x) (4.7)

The value of φ(x) between the data points φi (for x 6= i∆x), however, is not known and must be assumedby interpolation. The FDM uses linear interpolation to assume the shape of the continuous function betweenthe known samples. Interpolation means defining the values between discrete points as an analytic function,using the values of the points. Linear interpolation accomplishes this by using first-order linear functions:simple lines. An example of linear interpolation between data points is displayed in Figure 4.4a.

Linear interpolation of the discrete points φi allows the calculation of the local derivative: dφdx . Since the

interpolated piece is a simple line, its derivative is a constant gradient, defined by:

dφ

dx≈ ∆φi

∆x(4.8)

where ∆φi =φi+1 −φi , or the difference between two data points.

The Poisson equation requires a second derivative (see Equation 4.4). That implies the derivative of thederivative, or the change in change. The FDM must thus be applied again, but this time calculating the dif-ference between the gradients of two line pieces. This is also illustrated in Figure 4.4a. The second derivativeat φi is thus calculated as:

d2

dx2φ= d

dx

(dφ

dx

)≈

(∆φi∆x

)−

(∆φi−1∆x

)∆x

(4.9)


(a) Linear interpolation and the Finite Difference Method. The samples φi(red) are interpolated with first-order linear functions between every twopoints. The second derivative at the blue point is determined with the FDMby calculating the difference between the gradients ∆φi and ∆φi−1.

(b) Expanding the FDM to a data point φi , j ,kin 3D space.

Expanding ∆φi =φi+1 −φi , as displayed in Figure 4.4a, Equation 4.9 collapses to:

d2

dx2φ≈ 1

∆x2

(φi−1 +φi+1 −2φi

)(4.10)

This same process can be applied to the full left side of the Poisson equation in Equation 4.4, expandingthe FDM to three dimensions, with in the x, y and z directions respectively Nx , Ny and Nz data points. Thedata points indices are now i , j and k: φi , j ,k for i ∈ [1,2,3, . . . , Nx ], j ∈ [1,2,3, . . . , Ny ] and k ∈ [1,2,3, . . . , Nz ]. Ascan also be seen on the left side of Equation 4.4, the Laplace operator is the sum of three second derivativesover the three dimensions x (i ), y ( j ) and z (k). All points are still equally spaced with a distance of ∆x, as isdisplayed in Figure 4.4b, but a single data point φi , j ,k now has not 2, but 6 neighbors. Applying Equation 4.9for a 3D space thus produces:

∇2φ≈ 1

∆x2

(φi−1, j ,k +φi , j−1,k +φi , j ,k−1 +φi+1, j ,k +φi , j+1,k +φi , j ,k+1 −6φi , j ,k

)(4.11)

In this way, the FDM can be used to define the left side of the Poisson equation, if the given function f(and therefore the sought functionφ) consists of discrete points. Substituting Equation 4.11 into Equation 4.1and shifting some variables gives:

6φi , j ,k −φi−1, j ,k −φi , j−1,k −φi , j ,k−1 −φi+1, j ,k −φi , j+1,k −φi , j ,k+1 =−∆x fi , j ,k (4.12)

This equation is known as the discrete 3D Poisson equation for uniform spatialization (all points are dis-tanced equally). When f and ∆x are given, the values of φ can be found by solving all Nx ×Ny ×Nz linearequations. Solving these equations is the basis for the Poisson solver, which is done by applying the Conju-gate Gradient method, a method used to find the factor vector in a matrix-vector product.

Conjugate Gradient methodBefore applying the Conjugate Gradient (CG) method, the discrete Poisson equation and all its factors mustfirst be rewritten. The system of Nx × Ny × Nz = Ne linear equations can be interpreted as a Matrix-VectorProduct (MVP) by “flattening” the 3D samples space into column vectors,φ and b:

φ=[φ1,1,1,φ2,1,1, · · · ,φNx ,1,1,φ1,2,1, · · · ,φNx ,Ny ,1,φ1,1,2, · · · ,φNx ,Ny ,Nz

]T(4.13)

b =−∆x[

f1,1,1, f2,1,1, · · · , fNx ,1,1, f1,2,1, · · · , fNx ,Ny ,1, f1,1,2, · · · , fNx ,Ny ,Nz

]T(4.14)

(Note that the 3D samples have been flattened such that x-neighbors are 1 index apart, y-neighbors areNx indices apart, and z-neighbors are Nx ×Ny indices apart. This is how MATLAB stores 3D arrays. Any otherconsistent ordering of the 3D elements is also possible.)

The MVP associated with the discrete Poisson equation then becomes:

4.3. Solving the discrete Poisson equation in 3D 33

Aφ= b (4.15)

The matrix A holds the factors of the elements of φ on the left side of the discrete Poisson equation. It iscalled the Laplacian matrix, because it represents the Laplace operator ∇2. The Laplacian matrix defines theconnections in a 3D graph and thus describes which points in the list neighbor each other. Almost all of theentries in the Laplacian matrix are zero, because just a small amount of elements of the total Ne play a rolein a single linear equation, as is evident from Equation 4.12. This means that A is considered a sparse matrix.Figure 4.5 shows how a 3D graph translates to a Laplacian matrix.

Figure 4.5: An example of a Laplacian matrix for a 3D space of 5× 5 elements with ∆x = 1. The left shows the elements as they areconnected in a 3D graph, with their nodes colored according to their degrees. The right figure shows an image of the correspondingLaplacian matrix, with colors matching the 3D graph.

For non-uniform grids, the Laplacian matrix is not trivial and requires special methods to be constructed.An example is the more general form of FDM, called the Finite Element Method [2]. In any case however, theLaplacian matrix is a sparse matrix that describes how data points are geometrically connected in a 3D graph.For this work a uniform grid suffices, Equation 4.12 applies, and the values in A become trivial.

As can be seen in the right figure of Figure 4.5, the diagonal of A holds the degree of every element. Thedegree of a node in a graph represents how many other nodes are connected to it. For most of the elements,this number is therefore 6, but on edges of the model space, elements have less neighbors, and the degreecan be as small as 3 (for the 8 corners of the 3D space). Off-diagonal, every column that is connected to theelement index on its row, has the value -1. A consequence is that the matrix is by definition symmetrical(AT = A) and positive-definite (xT Ax > 0 for any vector x).

Now that the system of linear equations is defined as an MVP, it can be solved forφ. An obvious solution -especially within MATLAB, software tailored to matrix manipulation - would be to invert the Laplacian matrixA, and then multiply it with the right side of the equation, b, so thatφ= A−1b. Unfortunately, the matrix A isextremely large (for a 20×20×20 system, it has 64 million entries) so simply inverting the matrix is, even forMATLAB, expensive and unnecessarily cumbersome.

Instead, iterative methods can be used to attempt to fit a vector φ′ such that Aφ′ ≈ b. Iterative meth-ods accomplish this by repeated trial and error to try to minimize the residual vector r = ∣∣Aφ′−b

∣∣ until itsmagnitude |r | is acceptably low. One of these methods is the Conjugate Gradient method.

The CG method finds a set of conjugate vectors, and adds them together to form the solution φ. Everyiteration, a coefficient is calculated depending on the residual vector r , which determines the next conjugatevector. The vectors are then added until the residual vector is sufficiently small (or a max number of iterationswas reached). The specific algorithm associated with CG will be omitted here for brevity and can be found infull in the model source code [47].

Implementation and optimizationThe Laplacian matrix A is a sparse matrix, which means that almost all of its entries are 0. Therefore it isunnecessary, and resource-intensive, to save the entirety of the matrix for the lattice grid. Instead, only thenon-zero values of A are stored, accompanied by coordinates to show their row and column numbers. Thematrix is divided into rows, because the rows correspond with the elements in the lattice grid and the indices


of the lattice elements can serve as row coordinates. Every latticeElement object therefore has a fieldA_row, which consists of a two-column list of coordinates and their values. This reduces the size of a row ofA to 14 entries at most (1 node plus 6 edges, with coordinates) compared to 8000 entries for e.g. a 20×20×20system. Multiplication of a sparse matrix with a vector is still possible and relatively easy to implement: everyrow of the solution vector is equal to the sum of the entries in the sparse matrix, multiplied by the entries inthe row vector as indicated by the column coordinates in the sparse matrix.

The operator∇ (Equation 4.2) is also stored per row in every latticeElement. This operator is defined aseither the divergence operator, operating on a vector field to produce a scalar field, or the gradient operator,operating on a scalar field to produce a vector field. The latter is necessary to calculate the electric field vectorfor every lattice element using:

∇φ= F (4.16)

Since ∇ produces a vector from a scalar, it is discretely represented by three matrices, using the coeffi-cients of FDM (Equation 4.8). This makes it necessary to sparsify it, similar to A: instead of one entry percoordinate however, Nabla_row holds three.

This concludes the description of the implementation of the Poisson solver.

4.4. Calculating the current: the charge transport modelThe charge transport model is reproduced from the multiphonon-assisted Trap Assisted Tunneling (TAT)solver as presented in [59, 91]. This section will explain the specifics of both the physics and the softwareimplementation of the TAT solver.

First, some of the basics of semiconductor physics concerning TAT are explained. Then, electron captureand emission times of traps and their relevance are explained. After that, the phonon particle is explainedand its role in TAT. Then the density of states (N (E)) and the Fermi-Dirac occupation probability ( f (E)) [71],both factors in the capture and emission times, are explained.

Calculating the TAT current also requires calculating a tunneling probability, for which the Wentzel-Kramer-Brillouin (WKB) approximation [71] is used. The implementation of the WKB calculator is also explained in asubsection. The subsection after that concerns the calculation of phonon capture and emission rates.

The last two subsections explain how the electron capture and emission times of traps are applied tocalculate the current through the separate percolation paths. Then finally, the MATLAB implementation ofthe full TAT solver and its optimizations are described.

Basics of TAT semiconductor physicsThe theoretical physics of TAT require some basic understanding of semiconductor physics, specifically on:

• Crystal lattices and vacancies;

• Conduction and valence bands;

• The effect of vacancies on the conduction band.

On a molecular scale, every material has a certain structure. The chemical formula that describes thematerial, e.g. HfO2, only shows how much of each atom exists in the material, but its properties also relyon how these atoms are structured. The degree of structure in a solid material is defined by three crystallinephases as displayed in Figure 4.6.

• Amorphous: there is no or very little structure in the lattice, all the atoms are randomly connectedthrough the material.

• Poly-crystalline: separate grains of structured material, misaligned with each other.

• Crystalline: the entire material is structured.

The crystal structure is, however, never perfect and there are always some lattice defects. These are notnecessarily manufacturing defects, as they play a crucial role in the normal operation of RRAM, but they areparts of the lattice that deviate from the overall structure. Two types of lattice defects are relevant for RRAMoperation: vacancies and interstitial ions, displayed in Figure 4.7.

4.4. Calculating the current: the charge transport model 35

Figure 4.6: The three crystalline phases: (a) Amorphous, (b) Poly-crystalline and (c) Crystalline. Copied from [71].

Figure 4.7: On the left: a regular lattice, without lattice defects. In the middle: a missing atom leaves a positively charged vacancy. On theright: a negatively charged interstitial ion between the structured atoms.

• Vacancies are left by atoms that are missing in the lattice structure. If the vacancy in question is left byan oxygen ion - which is the case for metal oxides in RRAM - the vacancy is positively charged.

• Interstitial ions are extra ions that exist between the lattice structure. If the interstitial ion is an oxygenion, it is negatively charged.

Quantum physics dictates that an electron can only exist on certain discrete energy levels. This can beproven using the Pauli exclusion principle and Schrödinger’s wave equation [71], but this proof will be omit-ted for the sake of conciseness. However, it only applies to atoms which are spaced far apart. If the atoms arecloser together, the probability density functions of electrons, that exist around the atoms, start to interact.This splits the allowed energy levels into two. Adding another atom splits the energy levels again, and in thecase of many closely spaced atoms in a crystal lattice, the split energy levels are so numerous and so close to-gether that they form a quasi-continuous band. Figure 4.8 shows how an energy band is created by splittinga single energy level into a range by moving atoms closer together.

Figure 4.8: The creation of an energy band of allowed electron energies, by reducing the mutual distance between atoms in a lattice.Adapted from [71].


In this way, the allowed energy levels for electrons in a material span several bands. Two of these bands areof particular interest for semiconductors: the valence band and the conduction band. These are the bandsthat are closest to the Fermi level of the material, an energy level that is a unique property of a material. In thevalence band, electrons do not have enough energy to conduct current, but in the conduction band, electronscan move freely and thus act as charge carriers for a current.

There can be a gap between the conduction band and the valence band, called the band gap. The lengthof this gap, called the band gap energy, is also a unique property of a material and defines its conductivity.Electrons can not exist in this gap - however, they can jump over it if they have enough energy. The threecategories of conduction are distinguished and displayed in Figure 4.9.

Figure 4.9: Three different kinds of materials: on the left, an insulator (e.g. a metal oxide). On the right, a conductor (e.g. a metal), in themiddle, a semiconductor.

If the band gap is large, it takes more effort for an electron to jump to the conduction band. Materials witha large band gap are therefore insulators and are used when minimal conductivity is desired. If a material hasno band gap, it takes hardly any effort for electrons to move around. This type of material is a conductorand is used when maximal conductivity is desired. The middle ground, a moderately large band gap, is thesemiconductor. Semiconductors can be used to control the conductivity of a material by applying a voltage,thus creating a “transfer resistor” - or as we know it, a transistor.

A metal oxide normally acts as an insulator and therefore used in semiconductor technology to separatedifferent conducting parts from each other. Figure 4.10 shows the energy bands of a metal-insulator-metalstack, like that of an RRAM device, when a voltage bias is applied to it.

Figure 4.10: Energy bands of an MIM stack with an applied voltage bias. The electron (orange) in the left electrode has to tunnel throughthe entire thickness of the oxide to reach the right electrode.


To get from the negatively charged electrode (the cathode) to the positively charged electrode (the anode)the electron must go through a potential wall, as the conduction energy of the oxide is too high. This process iscalled tunneling. In a perfect oxide, an electron has to tunnel from the anode directly to the cathode, a processcalled direct tunneling or DT. For sufficiently thick oxides, as is the case in RRAM devices, this probability isnegligibly small, resulting in a very large resistance, as is expected of an insulator. However, as stated before,no crystal structure is perfect and some lattice defects will exist in the oxide. These defects aid the electron inits tunneling process.

Vacancies, specifically oxygen vacancies, act like positively charged points in the oxide. The positivecharge attracts the negatively charged electrons and thus an electron can be captured or “trapped” by thevacancy, as displayed in Figure 4.11. This trapping process is why oxygen vacancies in a metal oxide are alsoreferred to as traps - hence, trap assisted tunneling.

Figure 4.11: The process of electron trapping in an oxygen vacancy. The electron tunnels towards the trap and gets captured by it. Theelectron then emits from the trap to tunnel further.

In terms of energy bands, a trap is a local dip in the conduction energy of the oxide. This dip pulls theconduction band of the oxide downwards and thus creates an intermediate stepping stone for the electron totunnel towards. This is visualized in Figure 4.12.

Figure 4.12: Energy bands of an MIM stack with an applied voltage bias and traps distributed over the oxide. The electron (orange) nowonly has to tunnel through thin parts of the oxide to reach the anode.

Because the electrons now have smaller barriers to tunnel through, the traps increase the flow rate ofelectrons through the oxide and therefore the current. It is the dominant conduction method through anRRAM device when it is in HRS.

In the reference model by Padovani et al., the charge transport model is based on the TAT solver presentedby Vandelli et al. and Larcher ([59, 91]). This solver does not only account for the energy dip caused by pos-itively charged traps, but also for the effect of temperature on the energy of the electrons tunneling acrossthe traps. The temperature is related to discrete energy increases caused by multiple quasiparticles calledphonons. Thus these works simulate multiphonon-assisted TAT.


Phonons and their role in TATPhonons are a quasiparticle, which means that they act like particles, but do not physically manifest as aparticle. Their name comes from the Greek word φωνη (phonè), which means “sound”. Phonons representa wave travelling through a material, like a sound wave or any acoustic vibration. For example, hitting atable surface will make vibrations travel through the table, a process that can be represented by phononquasiparticles bouncing around inside the table.

These are acoustic phonons, but there are also optical phonons. This type of phonon does not representacoustic vibrations but rather vibrations caused by heat. Phonons are described by the frequency of the wavethey represent. The denomination “optical” comes from the fact that radiated heat is, in fact, light. When aheat source heats up a material, the energy from the source causes the surrounding atoms to vibrate in theirlattice positions. A heat source is therefore actually emitting phonons.

Phonons carry some energy with them, represented by the vibrating of the atoms that accompaniesphonons. The vibrating frequency of a phonon translates directly to a discrete energy level (for optical phonons)through Planck’s constant: Eph =ħωph . Traps are able to absorb this energy, giving an electron that is trappedinside the trap a “boost” to tunnel through the potential wall to the next the trap or electrode. Figure 4.13shows the effect this has on the energy bands of the TAT process.

Figure 4.13: Energy bands of the process of multiphonon-assisted TAT: electrons (orange), captured by a trap, receive a boost in theirenergy level attributed to the absorption of phonons (red) by the trap. The energy surplus of electrons captured by a trap is turned intophonons, which in turn increase the temperature around a trap (red gradient).

When an electron is captured by a trap, it was emitted before from a higher energy level. This means thatis has a surplus energy that is lost by the electron “falling” into the trap. The surplus energy is dissipatedas heat by the emission of phonons into the surrounding lattice. Traps are therefore heat sources; and thetemperature increase around the trap is determined by Fourier’s heat equation (Equation 4.6) which is solvedby the Poisson solver (Section 4.3).

Capture and emission timesThe flow rate of electrons in the TAT process is determined by the time it takes for a trap to capture and emitan electron. The shorter these times, the more electrons can pass through a trap over time, which is directlytranslated to an increase in current. The capture and emission times relate to the current as such: an electronflow rate Ri can be defined for every trap i that exists in the oxide, by looking at how many electrons the trapcan both capture and emit at a time. This is represented by the capture and emission rates of electrons fora trap i, (Rc,i , Re,i ) which in turn rely on the capture and emission times (τc,i , τe,i ), and the probability of anelectron occupying a trap ( ft ,i ) [59]:

Rc,i = τ−1c,i · (1− ft ,i ) (4.17)

Re,i = τ−1e,i · ft ,i (4.18)

Equation 4.17 implies that the capture rate relies on the time it takes to capture an electron, and theprobability that the trap is unoccupied (1− ft ,i ). Similarly, Equation 4.18 implies that the emission rate relieson the time it takes to emit an electron, and the probability that the trap is occupied ( ft ,i ).

The TAT solver [59] assumes that no charge buildup can occur in the traps. This means that the in-flow ofelectrons (capture rate) must be equal to the out-flow (emission rate). Since this means that Rc,i and Re,i areequal, the electron flow rate through a trap i can be defined as Ri = Rc,i = Re,i . Setting Equation 4.17 equal to


Equation 4.18 then defines the electron occupation probability of a trap by the capture and emission timesas:

ft ,i = τe,i ·(τc,i +τe,i

)−1 (4.19)

Inputting this equation in either one of the capture and emission rate equations, the electron flow rate for atrap i , Ri , becomes defined by only the capture and emission times [59]:

Ri = 1

τc,i +τe,i(4.20)

Equation 4.20 defines an electron flow rate for every trap in the lattice, which can then be used to findhow electrons percolate the the oxide through multiple percolation paths, as will be explained later in thissection. First, the calculation of the capture and emission times is explained.

The assistance of phonons is taken into account by summing over the (inverted) electron capture andemission times for all discretized energy levels caused by phonons [59, 91]. It is assumed the phonons in-volved are optical phonons of a single frequency, ω0. This means that a single phonon carries an energyof E = ħω0. Given m phonons are involved in the capture an electron, and n phonons are involved in theemission of an electron, the capture and emission times are defined by [59, 91]:

τ−1c,i =

∑m

(τc,i ,m

)−1 (4.21)

τ−1e,i =

∑n

(τe,i ,n

)−1 (4.22)

Where τc,i ,m is the time required for the capture of an electron in trap i, associated with the release ofm phonons (energy mħω0) into the surrounding lattice, and τe,i ,n is the time required for the emission of anelectron from trap i, associated with the absorption of n phonons (energy nħω0) from the surrounding lattice.τc,i ,m and τe,i ,n are defined as:(

τc,i ,m)−1 = Ni−1

(Ei ,m

)fi−1

(Ei ,m

)C ai ,mPT

(Ei−1,Ei ,m

)(4.23)

(τe,i ,m

)−1 = Ni+1(Ei ,n

)(1− fi+1

(Ei ,n

))Emi ,nPT

(Ei ,n ,Ei+1

)(4.24)

The factors of these equations represent:

• Ei : the energy level of an electron at trap i , determined by the local electric potential φi , the electroncharge −q and the trap energy Et ,i , as: Ei =−qφi −Et ,i

• Ei ,k : the energy level of trap i , with the addition of k phonons: Ei ,k = Ei +kħω0.

• Ni (E ): the density of states, in trap i , at energy level E .

• fi (E ): the Fermi-Dirac occupation probability, in trap i , at energy level E .

• C ai ,m and Emi ,n : the capture and emission rates that account for carrier-phonon interaction.

• PT(Ei ,n , Ei+1

): the tunneling probability, for an electron at energy Ei ,n towards a trap with energy Ei+1.

The following three sections will explain the calculation of the density of states (Ni (E)) and the Fermi-Dirac occupation probability ( fi (E)), the calculation of the tunneling probability (PT

(Ei ,n ,Ei+1

)) through the

WKB approximation and the formulae for the phonon capture (C ai ,m) and emission (Emi ,n) rates.

Density of states and Fermi-Dirac occupation probabilityFrom the first subsection of this section, it became clear that energy bands are not completely continuous,but rather consist of a very large amount of discrete energy states. The density of these states depends onthe difference from the conduction energy and reflects how many spaces there are for an electron to possiblyexist [71]. In [59], the density of states for an energy E is given as:

Ni (E) = 1

2π2

(2me

ħ2

)3/2 √E −Ei ·u (E −Ei ) (4.25)


Where me is the effective electron mass (a constant dependent on material properties), ħ is the modifiedPlanck constant, Ei is the conduction energy at trap i and u is the unit step function:

u (E −Ei ) ={

0 (E < Ei )

1 (E ≥ Ei )(4.26)

which reflects that in the forbidden region under the conduction band there are no allowed states.

While the density of states describes the amount of spaces that electrons can occupy, the Fermi-Diracoccupation probability describes the probability that these spaces are occupied. It depends on the differencefrom the Fermi level and is defined as [71]:

fi (E) = 1/

(1+exp

[E −EF,i

kTi

])(4.27)

Where EF,i is the Fermi level at trap i , k is the Boltzmann constant and Ti is the temperature at trap i .

The Fermi level of HfO2 used in the TAT solver is not given in any of the reference works [58, 59, 76, 89,91, 92] so it had to be deducted for this work, from the band gap energy Eg of HfO2. The band gap energy hasbeen measured [14] and found to be Eg = 5.8eV, but an assumption still has to be made for where the Fermilevel is within the band gap.

It is known that materials with a surplus of positive charge carriers have a Fermi level close to the conduc-tion band [71]. If the Fermi level is therefore assumed to be exactly in the middle of the band gap, with respectto the conduction band without energy dips caused by the traps, then locally on trap sites the Fermi level willbe much closer to the conduction band, as was already visible in Figure 4.10 and Figure 4.12. This indicatesthat the oxide locally has a surplus of positive charge, which further motivates the assumption. Therefore theFermi level is assumed to be constant relative to the trap-less conduction band of the oxide.

With this assumption in mind, the Fermi level at trap i becomes easy to calculate, knowing the localelectric potential φi and the width of the band gap Eg :

EF,i =−qφi −Eg /2 (4.28)

The product of the density of states and the Fermi-Dirac occupation probability reflects the amount ofelectrons that are present on a certain energy level, or the electron supply. Ni (E), fi (E) and their product areplotted in Figure 4.14, using the equations presented in this subsection and the parameters used in the TATsolver, all of which can be found in Appendix A. The figure shows that, near the conduction energy, the supplyof electrons is largest. Above that it exponentially decreases to 0. Under the conduction band, there are noelectrons, because levels below the conduction energy exists inside the forbidden band gap.

Phonon capture and emission ratesCharge carriers in the TAT process are assisted by multiple phonons at once, which can be captured andemitted from the trap at respectively a capture and emission rate. The factors C ai ,m and Emi ,n are calculatedin [59, 91] by the following given formulas:

C ai ,m = c0L(m) (4.29)

Emi ,n = c0L(n)exp

[−nħω0

kT

](4.30)

Where c0 is a variable dependent on the electric field magnitude F and the trap capture radius rt [59]:

c0 =4π2r 2

t

Eg

q2F 2ħ2me

(4.31)

Here q is the elementary charge, me is the effective electron mass and Eg is the oxide band gap energy.L(m) - in Equation 4.29 and Equation 4.30 - is the multiphonon transition probability for m phonons. It iscalculated by summing over the overlap of multiple phonon vibrational states between the trap with andwithout a trapped charge [91]. A visualization of this sum is displayed in Figure 4.15. This produces theformula:


Figure 4.14: Plots of the density of states (DOS) (top left), the Fermi-Dirac occupation probability (top right) and the electron supplyfunction (bottom). The energy on the x-axes is the energy relative to the Fermi level. The conduction energy at the trap site (Ei ) isdisplayed in orange.

Figure 4.15: Summing the overlap of vibrational states between a trap without charge (left) and a trap with a captured charge (right).Copied from [91].

L(m) =(

fB +1

fB

)m/2

exp−S(2 fB +1

)Im

(2S

√fB

(fB +1

))(4.32)

where fB is the Bose function:


fB = 1

exp[ħω0

kT −1] (4.33)

and Im is the modified Bessel function of the order m. This function is already implemented in MATLABby the function besseli. S is the Huang-Rhys factor, which represents the number of phonons requiredfor the rearrangement of the lattice around a trap to accommodate the trapped charge [91]. The value of Sdetermines the dependability on temperature of the TAT current. The values for these physical properties areprovided in a complete list in Appendix A.

WKB tunneling probabilityFinally, the calculation of the tunneling probability PT

(Ei ,n ,Ei+1

)by using the Wentzel-Kramers-Brillouin

approximation requires a dedicated implementation for this model. A separate function within the TAT mod-ule was constructed, that automatically and analytically calculates the tunneling probability PT between twolatticeElements, using the elements’ properties as parameters. To understand the implementation of thisfunction, a deeper understanding of quantum physics is first required, specifically on probability densityfunctions and Schrödinger’s time-independent wave equation [71]:

∂2ψ(x)

∂x2 + 2me

ħ2 [E −U (x)]ψ(x) = 0. (4.34)

Schrödinger’s time-independent wave equation is a second-order differential equation that is used tofind the wave function ψ(x), which describes the time-independent behaviour of a particle in a space. Thewave function is mathematically complex and is based on some physical constants (me , ħ), the energy of theparticle involved (E) and the local potential energy (U (x)). The local potential energy, in this case, representsthe conduction energy in the energy bands diagram. The magnitude squared of the wave function produces

the probability density function:∣∣ψ(x)

∣∣2. This function gives the probability that a particle exists on certainpoint x in space.

Based on the difference between E and U (x), two regions can be distinguished: the classical region andthe quantum tunneling region. In the classical region, where E >U (x), the wave function looks like a sinu-soidal function that bounces off potential walls, as is expected according to classical theory. However, in thequantum tunneling region, where E <U (x), the wave function describes the particle existing inside the po-tential wall. This is impossible within the limitations of classical physics, but quantum physics has shownthat tunneling is definitely possible and that it is a major contributor to charge transport models, such as TAT.

Figure 4.16 shows the solution to Equation 4.34 for a particle of E = 1eV, incident on a potential wall ofUw all = 4eV of tw all = 3Å thick, starting at position x = 0. The probability density function inside of thepotential wall takes the shape of an exponential function. To the right of the wall, the probability density isstill > 0, implying that it is indeed possible that the particle tunnels through the wall.

The WKB approximation is a method of predicting the shape of ψ(x) and approximating it by neglectingrelatively insignificant factors. This saves effort calculating the tunneling probability, which is a necessity:because many traps may exist in the oxide, the calculation of the flow rate between two traps may be calledmillions of times per simulated time step. The WKB approximation method assumes that the solution ofSchrödinger’s wave equation is a complex exponential function shaped like:

ψ(x) = exp[Φ(x)] (4.35)

where the first derivative of the exponent,Φ′(x), takes the shape of

Φ′(x) = A(x)+ j B(x) (4.36)

The approximation concerns the values of A(x) and B(x) inΦ′(x). A(x) is the real part ofΦ′(x) and there-fore translates to the change in magnitude of the wave function: |ψ(x)|′. B(x) is the imaginary part of Φ′(x)and therefore translates to the change in phase of the wave function: ∠ψ(x)′.

The magnitude of the wave function is the most interesting, because it is linked directly to the probabilitydensity function, which is used for the calculation of tunneling probability PT . Because only tunneling isconsidered, E <U (x) is assumed. This leads to an approximation possibility: as can be seen in Figure 4.16,


Figure 4.16: The solution to Schrödinger’s time independent wave equation for a particle incident on a 3Å potential wall, solved withhelp of MATLAB’s Symbolic toolbox. The crossed box indicates the potential wall. The blue parts represent the classical region (ψ f r ee (x))

and the red parts the quantum tunneling region ψtunnel (x). The bottom plot shows the probability density function,∣∣ψ(x)

∣∣2.

in the quantum tunneling region, the phase of the wave function changes very little. This can also be provedmathematically [71], but is omitted here for the sake of conciseness.

If the phase change is insignificant, as is the case in the quantum tunneling region, it can be assumed thatB(x) ≈ 0. This setsΦ′(x) = A(x) and the probability density function becomes:

∣∣ψ(x)∣∣2 = exp

[2∫ x

x0

A(x ′)dx ′]

(4.37)

where x0 is the turning point between the classical and non-classical regions. Using the 0th-order WKBapproximation for quantum tunneling [71]:

A(x) = A0(x) =−√

2me

ħ2 (U (x)−E) (4.38)

The tunneling probability is then retrieved by calculating the quotient of the probability density functionat the end and at the start of the potential barrier [71]. A particle tunneling from trap i to trap j , meeting apotential wall between xi and x j , therefore has a tunneling probability of:


PT =∣∣ψ(x j )

∣∣2∣∣ψ(xi )∣∣2 =

exp[

2∫ x j

x0A(x ′)dx ′

]exp

[2∫ xi

x0A(x ′)dx ′

] = exp

[−2

p2me

ħ∫ x j

xi

√U (x ′)−Edx ′

](4.39)

Now, if U (x) is given as an integrable function, it can be directly substituted in Equation 4.39 to analyticallyderive the tunneling probability. This is faster than an iterative function, because after analytical derivation, ittakes just a single computation to calculate PT . Fortunately, the potential at the start (Ui ) and the end (U j ) ofthe potential barrier is already known from the solution of the potential field provided by the Poisson solver.It is therefore possible to define the potential U (x) between two traps distanced by Di , j as a simple first-orderlinear function, as displayed on the left of Figure 4.17:

U (x) =Ui −Fi , j x (4.40)

where Fi , j =(Ui −U j

)/Di , j .

However, a single line does not properly reflect the valley-shaped potential dip in the conduction energy,as previously displayed in Figure 4.12. Larcher [59] proposes to shape the potential barrier instead as a three-piece linear function. The traps have a capture radius (rt ,i ) that reduces the potential barrier around the twotraps. The function U (x) now consists of three regions:

U (x) =

U1(x) = Urt ,i

rt ,ix

(x < rt ,i

)U2(x) =Ui −Fi , j x

(x ≥ rt ,i

)∪ (x < Di , j − rt , j

)U3(x) =U j −Et , j + Et , j +Fi , j rt , j

rt , j

(Di , j −x

) (x ≥ Di , j − rt , j

) (4.41)

where Urt ,i =U (rt ,i ), Urt , j =U (rt , j ) and Et , j is the trap energy of trap j . The function is displayed on theright side of Figure 4.17. Note: the zero potential point (U (x) = 0) was chosen to be on the bottom of trap i(Ui = Et ,i ) (see also the black dashed line in Figure 4.17). The zero could be placed anywhere, as Equation 4.39depends only on the difference between the particle energy and the potential barrier (U (x)−E). This choicewas made so that E can be defined as the difference between the particle energy and the bottom of trap i .

Figure 4.17: On the left: the potential function U (x) between trap i and trap j , using a single linear function over the mutual distanceDi , j . On the right: the potential function U (x) in three linear functions, their regions determined by the trap capture radii, rt ,i and rt , j .

An effect of the trap capture radii is that once the mutual distance between traps decreases, the captureradii start to overlap [59, 91]. This greatly reduces the potential barrier and thus increases the tunnelingprobability. At a certain critical mutual distance, the potential barrier vanishes completely, and the tunnelingprobability PT = 1 [76]. This means that now a defect subband is formed: electrons no longer have to tunnelto cross the oxide and drift current overtakes TAT as the dominant conduction mechanism. In this way, thereduction of the potential barrier is able to smoothly model the transition from TAT to drift current.


The MATLAB function that calculates the WKB tunneling probability, getTunnelingProbability, de-termines PT by picking a precalculated formula, analytically derived from Equation 4.39. Since only the po-tential barrier (U (x) > E) is considered, the integration limits must be adapted according to the level of theparticle energy E and the shape of U (x). The following list lists the four possible cases and their correspond-ing precalculated formulas. The cases are visualized in Figure 4.18.

Figure 4.18: The four possible cases of integration for the WKB tunneling probability. The blue shaded part of the potential function,where U (x) > E , represents the result of the integration, which is used in the exponent of Equation 4.39.

• Case 1: the particle energy E is above the potential barrier over the entire considered distance Di , j .Regardless of the shape of U (x),

PT = 1 (4.42)

• Case 2: the potential difference between the traps is so large, that the right-most part of the potentialfunction is completely below the particle energy E . U (x) is now a two-piece linear function of U1(x)and U2(x):

PT = exp

[4

3

p2me

ħ

(− rt ,i

Urt ,i

(Urt ,i −E)3/2 − 1

Fi , j(Urt ,i −E)3/2

)](4.43)

• Case 3: the mutual distance between the traps is so short, that the trap capture radii overlap. U (x)is now a two-piece linear function of U1(x) and U3(x). The crossing point xc , where U1(xc ) = U3(xc ),defines the boundary between the two regions:


xc =U j −Et , j +

(Et , j

rt , j+Fi , j

)Di , j

Uirt ,i

+ Et , j

rt , j

(4.44)

The tunneling probability is then:

PT = exp

4

3

p2me

ħ

− rt ,i

Urt ,i

(Urt ,i

rt ,ixc −E

)3/2

− 1Et , j

rt , j+Fi , j

(U j −Et , j +

(Et , j

rt , j+Fi , j

)· (Di , j −xc

)−E

)3/2

(4.45)

• Case 4: if none of the above three cases apply, then the potential function is equal to Equation 4.41.Integrating over the three regions of the potential barrier then gives:

PT = exp

4

3

p2me

ħ

− rt ,i

Urt ,i

(Urt ,i −E)3/2 + 1

Fi , j

((Urt , j −E)3/2 − (Urt ,i −E)3/2

)− 1

Et , j

rt , j+Fi , j

(Urt , j −E)3/2

(4.46)

Note that these options have multiple edge cases where the formulas that were listed have to be modifiedslightly, e.g. when Fi , j = 0, or when one side is an electrode and has no capture radius (rt ,i / j = 0). For con-ciseness, these edge cases have been left out of this section. The full implementation is documented in theMATLAB code on the TU Delft repository and does take into account all possible edge cases.

Percolation paths, power and chargeThe methods for calculating the electron capture and emission times τc,i and τe,i , together with all the factorsin Equation 4.23 and Equation 4.24, have now been explained. Now the electron flow rate Ri can be calculatedfor every trap i in a percolation path. Note, however, that Equations 4.23 and 4.24 concern not one, but threetraps: the current trap in the path (trap i ), the trap before this trap (trap i − 1) and the trap after this trap(i + 1). This implies that the electron flow rate through a trap depends both on the trap the electrons areflowing from, and the trap the electrons are flowing towards.

The process of finding percolation paths is based on the assumption that charge carriers will alwayschoose the path of least resistance [59]. Electrons start at the cathode (the negatively charged electrode)and jump from trap to trap, across the oxide, to the anode (the positively charged electrode). The path it takesin between is determined as such:

1. The electron starts at the cathode, and picks from all the traps the pair of traps with the highest ratefrom the cathode (cathode → trap → trap). If a single trap path has the highest rate (cathode → trap →anode) this path is picked instead. Any traps that were picked are marked as being in a path.

2. The rate from the current trap to the anode is calculated, together with the rates from the current trapto all other unmarked traps.

3. If the rate to the anode is the largest, or if no unmarked traps remain, the path of the electron ends as ithas reached the anode and the loop exits.

4. If the rate to another trap is largest, this trap is selected as the next one in the path and marked. Theloop returns to Step 2 and continues.

Every time the loop exits, the minimum rate in the calculated path is selected. This “bottleneck” rate Rmi n

determines the flow rate for the entire path, because electrons in the path can not flow faster than its weakestlink. The current through this percolation path is then calculated as such:

Ipath =−qRmi n (4.47)

Note that the charge of an electron is negative, resulting in a current direction from the anode to thecathode, as is expected.


The path finding process is repeated, until no more unmarked traps are in the oxide. The total currentthrough the device is then calculated by summing all individual path contributions:

Itot al =∑

Ipath (4.48)

The full process of finding percolation paths is illustrated in Figure 4.19.

Figure 4.19: The full process of exhaustively finding percolation paths to include contributions of all traps (blue) in the oxide (yellow).Marked traps are purple, bottleneck traps are red.

The electron flow rate of the full path is also used to calculate the power dissipation in each of the traps. Ifan electron jumps between a trap i and a trap j , it loses an amount of energy equal to ∆E = Ei −E j . This lostenergy is translated into dissipated power by using the flow rate of the path it is in:

Pi =∆E ·Rmi n (4.49)

Meanwhile, the capture and emission times define the probability of an electron being trapped in the trap( ft ,i ), according to Equation 4.19. Since the capture and emission times of every trap are already calculated,the electron occupation probability can be easily calculated to determine the net charge of the trap:

Qi = q · (1− ft ,i)

(4.50)

Note that the change in trap charge due to trapped electrons is not mentioned clearly in the referencemodel [76], nor does it clarify the charge of a trap without an electron occupying it. The microscopic modelby Duncan et al. ([27]) indicates however that, while different trap charges can exist, traps charged by +1ehave the most tendency to cluster together. Since this model concerns the formation of a filament caused bytrap clustering, the unoccupied trap charge was therefore set as +1e.


It was found that this strategy of finding percolation paths can sometimes cause a sudden decrease incurrent, even though the voltage is increasing. On further investigation it was discovered that the strategyhas a tendency of picking local maxima: a locally higher electron flow rate can cause the algorithm to pick adetour path which eventually ends up reducing the total current flow. This was fixed by storing the calculatedpaths of the previous run of the TAT solver and applying them whenever the calculated current is lower thanexpected.

Implementation and optimizationsIn the MATLAB implementation of the above described TAT solver, some optimizations were made to improvethe speed of the module. This subsection will address these implementation details and optimizations. Aflowchart of the TAT solver is displayed in Figure 4.20. The full implementation consists of the followingsteps:

1. From the provided lattice grid, extract the traps (latticeElements of type 'V+2') and sort them ac-cording to their potential φi . The reason for this will become clear below.

2. Decide which of the two electrodes is the cathode and which the anode, depending on the voltage atthe electrodes (types 'upper' and 'lower'). Then, using the function getRate, calculate all electronflow rates:

• From the cathode to a trap i to the anode;

• From the cathode to a trap i to a trap j ;

• From a trap i to a trap j to the anode.

Two optimizations were made for this step.

Firstly, every latticeElement has the index of the closest 'upper' and 'lower' electrode elementpre-stored. These indices are precalculated during the building of the lattice grid and, since the elec-trodes are considered immutable, they are valid throughout the entire simulation. Therefore, the com-putationally intensive process of finding several thousands of closest distances only has to be executedonce per a simulation.

Secondly, the rates between the traps and electrodes are similarly calculated only once outside the pathfinding loop and then stored, so that they can be checked for every step of the pathfinding loop withouthaving to calculate them again (see also Figure 4.20). Since the getRate function is computationallyintensive, minimizing calls to it is important to optimize the TAT solver.

3. A loop is started that exhaustively finds paths across traps between the two electrodes that works asfollows.

(a) Pick the largest trap rate from the cathode to an unmarked trap. A second loop is started that findstraps in the percolation path:

i. Mark this trap.

ii. Calculate all rates from this trap to unmarked traps. This is by far the most computationallyintensive process of the TAT solver: this action exists in a loop within a loop and concerns allthe traps in the lattice, possibly resulting in millions of calls to the getRate function per step.The computation time is improved greatly by assuming that electrons can only tunnel to-wards a trap of a lower electric potential. Since the traps were already sorted by potentialin the first step of the TAT solver, all traps with an index smaller than i are thus consideredmarked. This limits the getRate calls to only unmarked traps with indices above i .

iii. If the (precalculated) rate to the anode is larger than the rates to all other traps, exit this loop.Otherwise, repeat the trap finding loop.

(b) With the found path, calculate the path current, the power dissipated in every trap and the trapcharge according to the electron occupation probability.

(c) If there are no more unmarked traps left, exit this loop. Otherwise, repeat the path finding loop.

4. Finally, sum all the individual path contributions to the total device current, collect the dissipatedpower into the power profile P (x, y, z) and collect the trap charges into the charge profile Q(x, y, z).Return these values to the base model object.


Figure 4.20: The MATLAB implementation of the TAT solver as a flow chart. The submodules calculating the physics explained in theprevious subsections are yellow. The rate calculation submodule, the path finding loop and the trap finding loop are indicated by ashading with a dashed line. Calls to the rate calculation submodule have a thick outline, indicating their increased computationalintensity.


This concludes the description of the TAT solver. However, TAT is not the only charge transport modelthat can cause a flow of current through an RRAM device. For thin oxides (< 4nm [91]), the contribution ofelectrons tunneling directly from electrode to electrode also becomes relevant.

Direct tunnelingFor the calculation of Direct Tunneling (DT), the Tsu-Esaki model was used [35, 88]. This model calculatesthe net current density by an integration:

J = 4πme q

h3

∫ Emax

Emi n

PT (Ex ) N (Ex )dEx (4.51)

where me is the effective mass of an electron, q is the elementary charge, h is the Planck constant, andPT (Ex ) and Ns (Ex ) are respectively the tunneling probability and the supply function at energy level Ex :

• PT (Ex ) is calculated in the same way as in the TAT solver, by using the WKB approximation. The ap-proximation used is a simplified version of the previously explained module, since DT always concernstunneling from an electrode to an electrode, and thus involves no barrier reduction due to positivelycharged traps.

• Ns (Ex ) describes the difference in the supply of carriers at the interfaces of the dielectric:

Ns (Ex ) =∫ ∞

0

(f1 (E)− f2 (E)

)dEp (4.52)

where Ep is the longitudinal part of the energy level, parallel to the electrode. Filling in the Fermi-Diracoccupation probability for f (Equation 4.27) and integrating, gives:

Ns (Ex ) = kT log

1+exp[−Ex−E f ,1

kT

]1+exp

[−Ex−E f ,2

kT

] (4.53)

where k is the Boltzmann constant, T is the temperature, and E f ,1/2 is the Fermi level at either elec-trode 1 or 2.

The integration of Equation 4.51 is not trivial and therefore has to be performed by using a middle Rie-mann sum: for a number of discrete values within the limits Ex ∈ [Emi n ,Emax ], the value of the to be inte-grated formula in Equation 4.51 is calculated. Then the area between two discrete energy points Ex,i andEx,i+1, connected by a line, is calculated by averaging between the two points:

J = 4πme q

h3

Emax∑Ex,i=Emi n

1

2

(PT

(Ex,i

)N

(Ex,i

)Ex,i +PT

(Ex,i+1

)N

(Ex,i+1

)Ex,i+1

)(4.54)

The limits of integration Emi n and Emax are based on the three types of conduction that contribute to DT[35]:

• Electrons tunneling from the conduction band (ECB):

– Emi n is the highest conduction band edge of the two electrodes;

– Emax is the highest conduction band edge of the dielectric.

• Holes tunneling from the valence band (HVB):

– Emi n is the lowest valence band edge of the two electrodes;

– Emax is the lowest valence band edge of the dielectric.

The sign of the integration limits must be flipped since the valence band is on the opposite side of theFermi level.

• Electrons tunneling from the valence band (EVB):

4.5. Determining state change: the kMC engine 51

– Emi n is the lowest conduction band edge of the two electrodes;

– Emax is the highest valence band edge of the two electrodes.

At a sufficiently high potential difference, the valence band of one electrode can be partly aligned withthe conduction band of the other electrode. If this is the case, this conduction type should be takeninto account. For this work this will never happen, because the band gap energy of HfO2 is 5.8eV, andin normal operation, there will never be 5.8V applied to the device. For completeness, however, it wasincluded.

The complete Tsu-Esaki process is performed separately for every electrode point at the cathode-oxideinterface, tunneling towards the closest pre-calculated point at the anode. This is necessary because in adefective device, the interface of an electrode may not be perfectly planar, which means the oxide thicknessis not constant, and thus the DT current density can differ over the area of the device.

After calculating Equation 4.54 for all three conduction types and for every point of the cathode-oxideinterface, the separate contributions of every electrode lattice element point are summed and the total DTcurrent is returned.

Note that DT charge transport is negligible for thick oxides [91]. In Padovani et al., the oxide thicknessused is 10nm, which is sufficiently thick to ignore DT in favor of the more dominant TAT and drift current.For defect analysis, however, DT current becomes a significant contributor. Therefore, it was also included inthe this work’s charge transport model.

This concludes the description of the complete charge transport model. The full MATLAB implementa-tion can be found on the repository of the TU Delft [47].

4.5. Determining state change: the kMC engineThis section will explain the kinetic Monte Carlo (kMC) engine, which picks from a list of events to be executedand so regulates the evolution of the model state over time.

First, the basics of the kMC method is explained. Then the three types of events that can occur in themodel, as described in [76], will be listed and elaborated. Finally, the implementation and optimization de-tails of the kMC engine will be addressed.

Basics of the kinetic Monte Carlo methodThe kMC method, also called the dynamic Monte Carlo method, is a method of describing the change of thestate of a stochastic process in steps. Each step is associated with a different time step, hence, “kinetic” or“dynamic” MC.

Every step in a kMC process represents a change from the current state, to any of the possible next states.Every transition associated with these next states, called an event, has a corresponding rate, for example Rx

for an event x. Events that occur more often have higher rates and vice versa.

The rates of all N possible events in the system are calculated and then summed up to a total rate:

Rtot =N∑

x=1Rx (4.55)

The position of every event in this sum is remembered. Then, a random value is picked from a uniformdistribution u ∈ [0,1]. The position in the total rate, uRtot , is then used to find the corresponding event x bychecking:

Rx < uRtot < Rx+1 (4.56)

The total rate is used to calculate the time step associated with the execution of this event x:

∆t = R−1tot (4.57)

The kMC process is illustrated in Figure 4.21.Whether or not the chosen event x is executed is, in this model, based on a precalculated time step limit

provided by the model handler. If the total timestep ∆t is too large, the event will be discarded instead.


Figure 4.21: The basics of the kMC method, given five possible next states. The rates are summed up and an event is chosen by therandom value u.

Events in the modelThe model by Padovani et al. [76] uses three different kMC events to summarize the physical processes ofRRAM operation:

• Generation of a Frenkel pair: a Frenkel pair is a vacancy with an interstitial ion next to it. It thereforerepresents the generation of vacancies that form the filament, together with the ions that drift awaytowards the anode. Generation rates are calculated as:

GF (x, y, z) = ν ·exp

[−Ea −bF (x, y, z)

kT (x, y, z)

](4.58)

where ν= 7×1013 Hz is the effective vibration frequency of O-Hf bonds, Ea is the zero-field activationenergy to break O-Hf bonds, b = 40Å is the bond polarization factor, F (x, y, z) is the local electric field,k is the Boltzmann constant and T (x, y, z) is the local temperature.

The generation event rate GF (x, y, z) is calculated for every latticeElement that is an 'oxide' type,because these elements can turn into vacancies. However, neither the original work [76] nor the similarworks it references [58, 90] seem to address the directional nature of vacancy generation: if both avacancy and an ion are generated, where should the new ion be created? This model fixes this issue byturning the single event rate GF (x, y, z) into 6 seperate events, GF,−x (x, y, z), GF,−y (x, y, z), GF,−z (x, y, z),GF,+x (x, y, z), GF,+y (x, y, z) and GF,+z (x, y, z). Every one of these six events represents the correspondingoxide element turning into a vacancy type, with the oxide element at the indicated direction turninginto an interstitial ion type.

The rates are determined by projecting the local electric field F (x, y, z) onto the direction vectors point-ing towards where the ions will be pushed.

• Oxygen ion drift: an ion drifts from one interstitial position to another, pushed by the local electricfield. The reference model considers vacancies to be stationary [76]. Ion drift rates are calculated as:

4.5. Determining state change: the kMC engine 53

RD (x, y, z) = ν ·exp

[Ea,d −kD F (x, y, z)

kT (x, y, z)

](4.59)

where ν, F (x, y, z), k and T (x, y, z) are the same values as in Equation 4.58, but Ea,d is the activationenergy for an ion to drift to another position and kD is a field-induced barrier reduction factor.

The drift event rate RD (x, y, z) is calculated for every latticeElement that is an 'O−2' type, becausethese are the ions that can drift to a next position. Just like the vacancy generation rate, drift is of adirectional nature and the drift event rates are split into six events separated by the direction in whichthe ion will drift.

• Vacancy-ion recombination: if an ion exists next to a vacancy, it is possible that they recombine intoan oxide. Recombination rates are calculated as:

RR (x, y, z) = ν ·exp

[Ea,r

kT (x, y, z)

](4.60)

where ν, k and T (x, y, z) are again the same values as in Equation 4.58, but Ea,r is the activation energyfor an ion to recombine with a vacancy back into an oxide. This event is independent of the local electricfield.

The recombination event rate RR (x, y, z) is calculated for every latticeElement that is an 'O−2' typeand exists next to a 'V+2' type.

Implementation and optimizationsThe implementation of the kMC engine is, at its base, relatively simple. The calculation of the event rateshowever, is not. Even though the variables in the equations given in [76] are provided with the proper values,the resulting rates appear to be unrealistic. This will be further addressed in Section 5.3. For this section, itwill be assumed that the factors used in the calculation of the event rates are proper.

Before calculating the event rates, the provided grid of latticeElements is partitioned into two seper-ate grids: gridOxide, which holds all the 'oxide' elements, and gridIons, which holds all the 'O−2' ele-ments. The other elements types can not be the source of events, as generation is calculated only for oxides,and drift and recombination are calculated only for ions.

First, the Frenkel pair generation rates are calculated using Equation 4.58. The six rates per oxide elementover the six possible generation directions are calculated, and put in an Noxi de × 7 matrix, where Noxi de isthe amount of 'oxide' elements. The 7 columns store the index of the element and the six correspondingrates. If an event for a certain direction is impossible, for example when generation is blocked by a non-oxideelement, or the element exists on the edge of the model, the corresponding rate is set to 0.

After the oxide elements, the ion elements are considered. For every oxygen ion, the drift rates are calcu-lated according to Equation 4.59 and also the recombination rates according to Equation 4.60. Similar to theoxide rates, the ion rates are stored in a Ni on ×13 matrix. If it is impossible for an ion to drift in a direction,the rate is set to 0, and if there is no 'V+2' element neighboring on a certain direction, the rate is also set to0.

All the non-index entries of the rate matrices are then summed. Their position is still remembered, be-cause they are saved in two matrices. A random part of the total rate is determined using the built-in MATLABfunction rand, and finally the corresponding event is found by searching through the two rates matrices.

The event is returned from the kMC engine to be executed within the base model environment as theMATLAB structure event:

event = struct(...'type', 'G_F', ... The type of event, G_F, R_D or R_R'index', 3452, ... The index of the element where the event occurs'to', 6 ... The direction in which the event occurs (1 to 6 is −x,−y,−z,+x,+y,+z));


This event is then interpreted by the base model object and executed according to the descriptions givenin the previous subsection.

This concludes the description of the implementation of the kMC engine, and thus the description of thereference model. The following section will focus on the new addition to the model, the defect injector.

4.6. Injection of defects in the model deviceIn this final section on the model implementation, the new addition to the model will be introduced, thedefect injector. The function is used to interpret arguments from the model handler and then build a gridof latticeElements accordingly, as can be seen in Figure 4.2. This means that the defect injector functionentails both the building of the defect-free validation grid, and eventually the defective device grid for thedefect analysis.

First, the implementation of the defect-free grid builder will be described. Then, the implementation offour different kinds of defects will be explained.

Defect-free gridThe reference model [76] uses a MIM stack with a thickness of 10nm. The forming occurs at a grain bound-ary (GB) site [9] which is characterized by an increased density of pre-existing oxygen vacancies, due to themisalignment of the crystal grains as pictured in Figure 4.6b. Based on measurements of the leakage currentthrough an HfO2 stack [78], the vacancy density on the GB was set to 2.1×1021 cm−3, and the vacancy densityaround the GB, i.e. in the grain, was set to 1.9×1019 cm−3.

The reference model also uses top and bottom electrodes with a planar oxide interface. Because this workaims to include defects such as electrode roughness and otherwise misshaped electrodes, the electrode-oxideinterface can not be considered planar. Instead, the electrodes are modeled as a collection of grid points,which are in terms of implementation the same kind of object as oxide points (latticeElements, see alsoSection 4.2). The advantage of this implementation is that any electrode shape can be modeled, within theresolution of the lattice grid.

The spacing between the lattice elements is a point of discussion (see also Section 5.3). The referencemodel [76] and all of the earlier model iterations that it references itself [58, 90, 92] do not mention how thelattice elements are spaced inside the model. It is unknown whether the reference model even uses fixed,discrete positions (as in [1, 69, 82]), or rather a continuous space (as in microscopic models). Since the ratescalculated by the kMC engine rely on the existence of fixed event sites, that is the oxide and ion elements, itis assumed that the lattice elements must exist on fixed points in space.

The configuration is therefore assumed to be a regularly spaced grid, similar to other macroscopic RRAMmodels in the state of the art [1, 69, 82]. The grid distance between lattice elements was extracted from Figure4g-i in [76], together with images of the HfO2 crystal structure from [46], and was determined to be 3Å.

Because all the parts of the model are now points in space, they are a 3D graph, and can therefore besupplied with information for the Poisson solver (Section 4.3). Every lattice element i is supplied with afield A_row and Nabla_row, containing the entries of i -th row of respectively the Laplacian matrix A (Equa-tion 4.15) or the divergence operator matrix ∇ (Equation 4.2). Both are coordinate-encoded, to be compatiblewith the sparse-matrix-vector-product implementation of the Poisson solver.

The following list describes the steps to building the model lattice grid.

1. The electrode elements are set on the top and bottom edges of the grid, by checking limits imposed onthe z coordinate of the lattice elements.

2. After setting the electrodes, the remaining oxide is populated with vacancies assuming a uniform dis-tribution. Every 'oxide' lattice element has a probability of being a 'V+2' according to:

PV = (NV ·106)−1/3

(4.61)

Where NV is the vacancy density (in cm−3). NV can vary over different regions within the lattice, be-cause the vacancy density is higher around a grain boundary. If a vacancy is set, its energy Et ,i is chosenfrom a normal distribution, where the expected value µ= 2.8eV and the deviation is σ= 0.1eV [59].

4.6. Injection of defects in the model device 55

3. To ease calculation time in the TAT solver, for every element in the grid the shortest distance to thelower and upper electrode is calculated. The index belonging to this electrode point, that exists closestto the element in question, is also saved as a field in latticeElement.

4. Any points of the top electrode that directly border the oxide are set as interface layers, representing theoxygen gettering layer.

Now a grid oflatticeElementobjects is created. It was displayed using the base model functionplotGrid(Section 4.2) and displayed in Figure 4.22.

Figure 4.22: The grid of latticeElement objects for a defect-free RRAM device. The lower electrode elements are grey triangles(pointing up), the upper electrode elements are transparent triangles (pointing down), the interface layer elements are light grey dots,and the vacancies are white circles.

Apart from generating a defect-free grid, the defect injector is also able to add the three different types ofdefects described in Section 2.4 to the initial grid: oxide thickness variation, electrode roughness and impu-rities. These will be explained next.

Injecting oxide thickness variationVarying the oxide thickness is relatively simple: the thickness of the oxide is directly connected to the thick-ness of the entire lattice grid (including electrodes). Therefore, by varying Nz (the number of lattice elementsin the z direction) the oxide thickness of the model can be manipulated, as is shown in the lattice grids gen-erated and plotted in Figure 4.23.

Injecting electrode roughnessIntroducing electrode roughness requires some more implementation. An argument is added in the handlerthat defines the deviation from the regular, planar electrode surface. This deviation determines how manyextra electrode elements are added at the top and bottom electrodes by choosing a uniformly distributedrandom number between 0 and the provided electrode roughness argument. This fits the implementationof other models on the scale of this work [16, 80]. Grids generated with electrode roughness are displayed inFigure 4.24.


(a) Nz = 30 (tox = 7.2nm) (b) Nz = 50 (tox = 13.2nm)

Figure 4.23: Injecting oxide thickness variation in the lattice element grid. The nominal value for Nz is 40 (tox = 10.2nm). The legend isthe same as in Figure 4.22

(a) 1 element (σ= 0.3nm) (b) 4 elements (σ= 1.2nm)

(c) Plane cut-out throughmiddle of Subfigure (b).Different colors representdifferent element types,with dark blue: oxide, blue:lower electrode, light blue:upper electrode, cyan:interface layer, yellow:vacancy.

Figure 4.24: Injecting electrode roughness in the lattice element grid. Nominally, there is no electrode roughness. The legend is the sameas in Figure 4.22. Subfigure (c) shows a plane cut-out of Subfigure (b).

Injecting impuritiesAnother advantage of defining the model lattice grid as a set of discrete points is that certain regions can beassigned different types. In this way, impurities can be injected everywhere into the MIM stack. Impuritiesare implemented as spheroid regions with a horizontal and vertical radius. An impurity is thus defined byfour variables, listed in Table 4.1:

Table 4.1: The four variables that define an impurity, and examples of the values that could be supplied to it.

Variable Description Example input

Type The type of the impurity. 'air'

Width The horizontal radius of the impurity. 4nmHeight The vertical radius of the impurity. 3nmPos The x, y , z coordinates of the center of the impurity. [4.5nm,4.5nm,8nm]

Note that for the purpose of defect analysis, the new latticeElement type 'air' was introduced. Thiselement can not change type and has the sole purpose of blocking conduction paths. Therefore it could also

4.6. Injection of defects in the model device 57

be interpreted as any other foreign insulator contaminating the MIM stack.

A latticeElement exists within the defined (spheroid) impurity region if the following equation is true:

(x −xi m)2 + (y − yi m

)2

w2

(z − zi m)2

h2 < 1 (4.62)

where x, y , z are the coordinates of the element, xi m , yi m , zi m are the coordinates of the center of the impu-rity, w is the horizontal radius and h is the vertical radius. This description of an impurity allows a wide rangeof unique impurity shapes, as is demonstrated in Figure 4.25.

(a) An air bubble in the middle of the oxide.(w = 4.0nm, h = 3.0nm)

(b) A punch-through of the upper electrode.(w = 1.2nm, h = 10nm)

(c) A bump on the bottom electrode.(w = 5.0nm, h = 2.0nm)

(d) An oxide punch-through with holes in the electrodes.(w = 2.4nm, h =∞)

Figure 4.25: Four unique impurities injected into the MIM stack. The legend is the same as in Figure 4.22, with the addition of air,displayed as red x-es.

With the defect injector tool, a wide range of defects can be injected into the model’s MIM stack, to re-search the direct consequences on the resistive switching operation of the RRAM device. Paired with the ref-erence model [76], this model could provide a valuable insight into the symptoms and limits of defects, basedon a direct connection between the physical processes associated with RS and the electrical characteristics atthe terminals of a device.

This concludes the chapter describing the implementation of this work’s MATLAB model.

5Validation of model

In this chapter, the results of the implementation discussed in Chapter 4 will be compared with verified datasuch as experimental measurements and known laws of physics. First, the results of the Poisson solver willbe verified in Section 5.1 using known laws of physics that are a direct consequence of the Poisson equation.Then, the TAT-solver will be verified using experimental data extracted from the works it is based on [91]in Section 5.2. Afterwards the kMC engine will be discussed, with as a most important point the importantinformation missing from the reference model [76] in Section 5.3. Finally the full defect-free model will bepresented in Section 5.4, which, like the kMC engine, was missing important information and underwentseveral adjustments in attempts to have it operate reliably. Simulations were performed on the TU Delft QCEcluster.

5.1. Validating the solution of the 3D Poisson equation solverTo validate the solution from the 3D Poisson equation solver, the solution vector φ must be shaped so thatthe Poisson equation holds:

∇2φ= f . (5.1)

Or, in its discrete form:

Aφ= b (5.2)

Verifying that this equation always holds after a run of the Poisson solver validates that it correctly calcu-lates either the electric potential or the temperature.

First however, the Laplacian matrix A should be verified. If A incorrectly represents ∇2, the discrete Equa-tion 5.2 will hold, but the original, analytic Equation 5.1 will not. To verify that A ≈∇2, the outcome functionf is set to 0, producing the Laplace equation:

∇2φ= 0 (5.3)

Analytical solutions to the Laplace equation exist given boundary regions for φ. One of these analyticalsolutions is that of a region bounded by two (infinite) parallel planes of different potential. For this model,that translates to the two electrodes when a voltage is applied to them. In this case, the electric field profile isexpected to be constant:

Fi =− (VU −VL)/tox k (5.4)

where VL (VU ) is the electric potential at the lower (upper) electrode, tox is the oxide thickness and k is theunit vector in the z direction. The corresponding potential profile is obtained by applying the gradient (∇)and is therefore expected to be:

φi =∇ (Fi ) =VL +F · zi (5.5)

where zi is the coordinate of element i parallel to the oxide thickness and F is the magnitude of the electricfield.

59

60 5. Validation of model

Note that FDM uses only the adjacent neighbors for calculating the derivative in a direction. This meansthat even though the Poisson solver outcome φi will have a constant increase, the derived magnitude of Fon to the boundary points will be exactly half of Equation 5.4. The reason behind it is this: away from theboundaries, an element sees the same slope on both sides. On boundaries, an element sees a slope only onone side. This anomaly will be taken into account when verifying the equation Equation 5.4. It has no effecton the operation of the model, since the boundary points exist outside the simulation space, the oxide.

Verifying the Laplacian matrix with an analytical solutionFigure 5.1 shows the results when comparing the solution of the Poisson solver for Aφ= 0 (Equation 5.3).

Figure 5.1: Validating the Laplacian matrix of the Discrete Poisson equation in 3D from the solver described in Section 4.3. In readingorder: (a) Layout of the grid; (b) electric field magnitude F calculated by the Poisson solver; (c) the relative error compared to Equa-tion 5.4; (d) the boundary points used in the Poisson solver; (e) electric potential φ calculated by the Poisson solver; and (f) the relativeerror compared to Equation 5.5.

It considers an oxide that is 37 elements thick (including 2 interface layers). This means that the distancebetween the two electrodes is 38×3Å = 11.4nm. A voltage of 1V is applied to the upper and lower electrodes.According to Equation 5.4, this should result in a constant field of F = 8.772×107 V/m and a linear functionof z (Equation 5.5) for the Poisson equation solution φ.

The solution is indeed valid: given the lattice grid as displayed in Figure 5.1(a), the maximum relative dif-ference between the solution of the Poisson solverφ Figure 5.1(e) and that of Equation 5.5 is only ≈ 5.6×10−6

Figure 5.1(f). It can thus be assumed that the Laplace matrix is determined correctly. The maximum differ-ence between the calculated electric field F Figure 5.1(b) and that of Equation 5.4 is ≈ 1.4×10−4 Figure 5.1(c),therefore the calculation of the gradient ofφ can also be considered accurate.

Validating the full potential of the Poisson solverNow that the Laplacian matrix A has been verified, the Poisson solver can be put to the test. A random gridof latticeElements was generated, with a high density of point charges of +1e existing inside the grid. Theelectrodes are no longer planar but rather irregular, with a bump added to the lower electrode. A voltage of 2Vis applied to the electrodes. Figure 5.2 shows the results of applying the Poisson solver to this grid to calculatethe potential profile.

It can be seen that the Conjugate Gradient method reduces the relative error∣∣Aφ−b

∣∣/φ down to ≈7.2×10−7. Several observations can be made from the found solution:

• The large amount of charge inside the oxide greatly increases the electric potential inside the oxide(Figure 5.2(b)).

5.2. Validating the calculated device current 61

Figure 5.2: Validating the solution to the Discrete Poisson equation in 3D from the solver described in Section 4.3. In reading order:Layout of the grid; electric potential φ calculated by the Poisson solver; the error from Equation 5.2; and the magnitude of the electricalfield.

• The magnitude of the electric field is large (Figure 5.2(d)) and points outwards (not visible in Fig-ure 5.2(d), but evident from the visible gradient in (Figure 5.2(b)).

• Near the bump on the lower electrode, the electric field is of a higher magnitude, as the oxide is thinner.

• The pairs of lighter and darker spots in Figure 5.2(d) represent the superposition of a point-chargeelectric field on an existing field: if a field points in a direction, and a point charge field is superposedon it, it will amplify the field on one side and dampen it on the other side. Hence, the lighter and darkerspots.

These observations align with the expected behaviour of electrostatics, and are therefore enough validationthat the Poisson solver is indeed able to find a solution φ to the Poisson equation in 3D.

5.2. Validating the calculated device currentFor the validation of the charge transport model, measurements of the leakage current through the hafniumoxide gate of a NMOS transistor from Vandelli et al. [91] were used. The TAT model from Larcher and Vandelliet al., as described in Section 4.4, was applied to a similar setup. The device was not set up using the latticebuilder from the main model (Section 4.6), but, since only the traps are considered in this case, it was set upusing a reduced version of the lattice builder. The source code for this reduced lattice builder can be found inthe TU Delft repository [47].

This reduced version only generates vacancy elements and assumes constant, planar electrodes, as is thecase in the reference. The vacancies are generated randomly and uniformly throughout the grid according toa given density. They are not connected by a Laplacian matrix, because in this reduced lattice grid with planarelectrodes the electric field F is constant over space and the potential is a simple function of the position ofa trap parallel to the oxide thickness, φ(z). The setup therefore only concerns the TAT solver, and leaves outthe Poisson solver and the kMC engine.

The measurements in [91] were performed on a 5nm thick HfO2 oxide, layered onto a 1.1nm thick SiO2

interface layer and a TiN electrode. The device area is very large (30×30µm2), to average out statistical fluc-


tuations. Current density measurements were then performed on 4 ambient temperatures: 50K, 200K, 300Kand 400K, over constant values from 0 to 2V.

Within the HfO2 layer, there is a uniform vacancy density of 4.5×1019 cm−3. The vacancy density withinthe interface layer, as well as its thickness without the TiN electrode, was not clear from the referenced work[91] and therefore was left out. Since the oxide is thinner than in the reference measurements, the simulatedcurrent density is therefore expected to be higher than the measured current density, but the exact differencewill be investigated.

Instead of simulating a single 30× 30µm device, a large number (Nsi m = 1000) of smaller devices (20×20nm) was simulated in parallel. The statistical fluctuations of the current density are in this way equallyaveraged out, with the additional benefit of faster simulation by parallelization. A complete overview of allthe properties used in the verification of the TAT solver is shown in Table 5.1.

Table 5.1: Overview of all the variables used in the verification of the TAT solver, as obtained from [91]. A complete list of all propertiesused in simulations can be found in Appendix A.

Variable Description Value Note

tox Oxide thickness 5nm Thicker in reference, but unclear valuedx ×dy Device area 20×20nm (×1000) Larger in reference, but statistical fluctu-

ations still mitigatedET Trap energy in HfO2 1.4−2.4eV

S Huang-Rhys factor in HfO2 17rt Trap capture radius 5.64Å Obtained from trap cross section σt =

1×10−14 cm2 by rt =pσt /π

NV Trap density in HfO2 4.5×1019 cm−3

Figure 5.3 shows the setup of the TAT solver verification. The results of the simulation are shown andcompared to the measurement data in Figure 5.4. The deviation between the simulated current density andthe measured current density is shown in Figure 5.5.

Figure 5.3: The grid setup of the TAT verification simulation.

To validate the percolation path finder described in Section 4.4, the percolation paths were visualized bylines between the electrodes in Figure 4.19. The paths in the figure show that the electrons indeed prefer apath of least resistance, with some fluctuations due to the randomized trap energies (see also Table 5.1).

Some traps are not part of a percolation path. These traps exist too far away from the cathode and othertraps, which means that any possible path towards the trap was either already marked by another, faster path,or the path has a rate of 0.

5.2. Validating the calculated device current 63

Figure 5.4: The results of TAT verification simulation. The measurement data are displayed by symbols connected by dashed lines. Thesimulation results are displayed by unbroken lines.

Figure 5.5: The relative deviation of the results from the measurement data.

It is evident from Figure 5.4 that the simulation results do not completely represent the measurementdata: the simulation current is higher. The reason for this is that the SiO2 interface layer described in [91]was not included in this setup (Figure 5.3), because the vacancy density and thickness of this layer were,unfortunately, not sufficiently clear. This causes the oxide to be slightly thinner, which is expected to increasethe electron flow rates: the paths that the electrons have to take are shorter, the electric field is higher and thepotential differences larger.


Figure 5.6: An example of percolation paths as determined through the oxide, for V = 1V.

The relative deviation from the measurement data is displayed in Figure 5.5. The deviation from the mea-sured data appears to decrease for higher temperature and higher voltages. The largest deviation is observedfor 50K, which is shown to decrease rapidly for low voltages in the measurements, but not in the simulations(Figure 5.4).

Even though the results deviate from the measured data, they do appear realistic. The deviation decreasesas the temperature increases, which is in line with the information given in Vandelli et al.: the conductivityof HfO2 traps is more temperature-dependent than the missing interface layer, due to the larger Huang-Rhysfactor (S = 6 for SiO2, S = 17 for HfO2). The Huang-Rhys factor represents the ability of charge carriers tocouple with phonons [91] (see also: Section 4.4). Because of this higher temperature dependency, HfO2 isexpected to dominate conduction for higher temperatures.

For the purpose of this model, it was therefore decided that the charge transport model is valid. In the fu-ture, more research might be necessary to completely validate the model, but with the available informationin [91], this was unfortunately not possible.

5.3. Validating the events that change the model stateThe kMC engine should pick an event from a large list of events calculated using the rates that were elaboratedin Section 4.5. This section provides a critical analysis of the event rates that drive the kMC engine, using thevalues for the relevant variables that were provided by Padovani et al. [76] and its references. It is foundthat these values can not produce realistic rates in this kMC implementation and that, evidently, additionalinformation is missing in the original sources [58, 59, 76, 90, 91].

Recall (Section 4.5) that the three events that occur during the simulation of the model are the stress-induced generation of a vacancy-ion pair (GF ), the drift of an oxygen ion from one interstitial position to thenext (RD ) and the recombination of a vacancy-ion pair (RR ). The corresponding equations for these rates are:

GF (x, y, z) = ν ·exp

[−Ea −bF (x, y, z)

kT (x, y, z)

](5.6)

RD (x, y, z) = ν ·exp

[Ea,d −kD F (x, y, z)

kT (x, y, z)

](5.7)

5.3. Validating the events that change the model state 65

RR (x, y, z) = ν ·exp

[Ea,r

kT (x, y, z)

](5.8)

where F (x, y, z) is the magnitude of the local electric field, and T (x, y, z) is the local temperature. Thevalues for the remaining physical constants in these equations were obtained either from Padovani et al. orany of its references, and are displayed in Table 5.2.

Property Description Value

ν Effective O-Hf bond vibration frequency 7×1013 Hz [58, 76]Ea Zero-field effective O-Hf bond breakage energy 2.9eV [75, 76]b Bond polarization factor 40eÅ [75, 76]Ea,d Ion diffusion activation energy 0.7eV [76]kD Factor depending on material properties −3eÅ [58]Ea,r Recombination activation energy 0.2eV [76]

Table 5.2: The variables in the kMC event rate equations, the description of their meaning, and their value, together with the reference(s)from which the value was obtained.

Even though, in the reference work, these values for the properties appear to produce realistic results,verified with experimental measurements, the rates that are calculated using Equations 5.6 to 5.8 appearunrealistic. In order to clarify this problem, the constant and variable parts of the equation must be identified.

Every property listed in Table 5.2 is a constant, as well as the Boltzmann constant k. This means thatthe only variable parts of the event rates equations are the local electric field magnitude F (x, y, z) and thelocal temperature T (x, y, z). These are the parts that change over the course of a simulation and thereforefully control the change in event rates. Several problems with the equations can be pointed out by plottingthe outcomes of the event rate equations versus the temperature and electric field magnitude, as is done inFigure 5.7.

From this figure, three problems are apparent:

• The electric field strength plays a more important role that stated. Padovani et al. states that the softdielectric breakdown during the forming of the filament is mostly driven by an increase in temperaturein the grain boundary. They also state that in the calculation of the electric field magnitude, the chargeof vacancies and ions in the lattice is taken into account. From the results of this work’s Poisson solver(see also Section 5.1), it appears that a point close to a charged point experiences a significant localfield, which can possibly be much more significant than the externally applied field. This has the effectof events firing independent of the externally applied voltage, which is unrealistic: the high local fieldmagnitude is only increased with the addition of newly generated vacancies, which further acceleratesgeneration, causing a full dielectric breakdown within a nanosecond of simulation time. However, thereferenced model does not sufficiently mention the relevance of the local electric field to justify itsimportance in the event rates equations. To illustrate this problem, the electric field magnitude at thelattice element neighboring a +2e point charge vacancy is calculated at 3Å, using Coulomb’s law:

F = 1

4πε

q

r 2 (5.9)

where ε= εr ·ε0 is the dielectric permittivity, q is the magnitude of the point charge and r is the distancefrom the point charge:

F = 1

4πε

+2e(3Å

)2 = 320MV/cm (5.10)

For the electrode-induced field to reach this magnitude in an oxide of 10 nm thick, 320 V needs to beapplied to the terminals of the device. Obviously, this magnitude of electric field is too large, whichcauses a runaway generation of vacancies regardless of the applied voltage, even when no voltage isapplied at all. This problem is pictured in Figure 5.8.

Note that Coulomb’s law does not precisely describe the electric field calculated by the Poisson solver,because the Poisson solver also takes into account the boundary conditions that are imposed on theelectric potential at the electrodes. The order of magnitude is, however, comparably large.


Figure 5.7: A surface plot of the three kMC event rates vs. the local temperature T (x, y, z) and the local electric field F (x, y, z). The z axisshows the natural log of the event rate.

• The drift rate is never dominant. The rate that describes the diffusion of oxygen ions (RD , displayedin green in Figure 5.7) is never higher than the two other rates, for all values of T (x, y, z) and F (x, y, z).For low values of F (x, y, z), RR dominates, which implies that an oxygen ion will recombine before itdrifts away. For high values of F (x, y, z), GF dominates, which implies that a new vacancy-ion pair willform before the ion drifts away. In other words, the event describing the diffusion of an oxygen ion isoddly rare. This is unrealistic because the drift of oxygen ions is the main mechanism behind resistiveswitching in OxRAM devices [97]: thus, the rate is expected to be more dominant.

• For high values of the electric field, the temperature dependency flips. As can be seen in Figure 5.7,the rates that are influenced by the local electric field (GF and RD ) mirror their dependency on temper-ature as soon as they cross a certain threshold. This threshold represents the crossing of the barrier setby the activation energy, e.g. for the generation rate:

Ea −bF (x, y, z) = 0 (5.11)

This implies a local electric field magnitude of 3.59MV/cm for generation, and 23.3MV/cm for driftrates. For an oxide of 10nm thick (as in the reference model), this translates to the same applied voltagevalues. This means that if the applied voltage crosses 3.56V, generation rates no longer favor the highesttemperature regions, but rather the lowest temperature regions - which is unrealistic. The referencesconcerning this model all mention temperature being the main driving force behind the formation ofa filament [10, 58, 76]. In other words, the high temperature caused by the increased conductivity of aforming filament should not slow down generation, but accelerate it.

An explanation for the temperature dependency flipping for high electric field magnitudes could be

5.3. Validating the events that change the model state 67

(a) The initial state of the grid (left) and the magnitude of the local electric field (right). The grid shows electrodes (downward/upward triangles) andvacancies (blue).

(b) The state of the grid after ≈ 5×10−15 s (left) and the magnitude of the local electric field (right). The grid shows electrodes (downward/upward triangles),vacancies (blue) and interstitial oxygen ions (red). The red arrows indicate the direction of the electric field at the position of the ions.

Figure 5.8: An example of the runaway generation of vacancies, when no voltage is applied to the electrodes (V = 0V). In the grid, newvacancies can be seen generating randomly around existing vacancies, because the local electric field magnitude at these points is toohigh.

that the local electric field magnitude should never cross 3.59MV/cm in the normal operation of themodel. But, as was apparent from the first point of this list, the local field magnitude is able to far crossthat value independent of the applied voltage.

The above three problems are directly evident from Figure 5.7. However, there is one more problem thatmay not be immediately visible.

• The generation activation energy (Ea ) is too high for the referenced timestep. Using the values fromTable 5.2, the generation rate can be calculated for a case where, according to the reference work, theself-accelerated soft breakdown should be started. According to Figure 5.9, a result copied from thereference model, this is when ≈ 2.5V is applied to the device, corresponding with an electric field mag-nitude of 2.5MV/cm for the referenced oxide thickness of 10nm. If the temperature is already relativelyhigh (500K), the generation rate (Equation 5.6) for an element becomes:

GF (x, y, z) = 7×1013 Hz ·exp

[−2.9eV−40eÅ ·2.5MV/cm

1.381×10−23 J/K ·500K

]= 5.023×10−6 Hz (5.12)

Considering that the total timestep of a single model cycle corresponds to the sum of all rates (seealso Section 4.5), one could assume that every single lattice grid point in the 30×30×40 grid is at thereferenced critical condition for a vacancy to form. This means that the total timestep of the modelbecomes:

∆t = 1

Rtot= 1

36000 ·5.023×10−6 Hz= 5.530s (5.13)


Figure 5.9: I-V curve of the forming process as simulated in [76]. The forming voltage lies around 2.5V, where generation rates shouldpeak. Copied from Padovani et al. [76]

Which implies that, under the (very generous) assumption that every single lattice element exists atthe critical vacancy forming conditions, it still takes several seconds for a single vacancy to form. Thismakes no sense considering that realistic forming times, both referenced in Padovani et al. [76] and inliterature (see also Chapter 2) are in the scale of a few microseconds.

A reason for this oddly low generation rate could be that the local electric field, and by that the chargeof non-oxide lattice elements, plays a quite significant role in the formation of new vacancies. Thecalculation and effect of charge trapping in vacancies is however hardly mentioned by the referenceworks [58, 59, 76, 91]. The only information that is given is that effects such as charge trapping are takeninto account when calculating the electric field strength - the value of this charge, and the strategies ofmodeling these effects, are not mentioned. Without the charge of vacancies and ions, the local electricfield can not be calculated, and as stated above, the generation rates are too low.

In conclusion, there are too many unexplained oddities in the referenced implementation of the kMCengine. These oddities should be further investigated, or perhaps redefined entirely, in future works. For thiswork, however, the kMC engine was not able to realistically simulate the change of the device state over time,for the reasons shown above. It was therefore decided that state change had to be excluded from this work’sfinal model.

5.4. Validating the full modelApart from validating the separate modules of the model, the complete model also needs to be validatedby comparing it to I-V plots and lattice layout plots from the reference work [76]. Unfortunately, it becameclear in Section 5.3 that the kMC engine is unable to operate as presented in the reference model, becausetoo much information is either unrealistic or missing. Outside of the kMC engine as well, many contextualbut crucial details remain unmentioned in the reference work. Modules such as the Poisson solver used, thespacing of the lattice elements, charge of the lattice elements, details of the kMC engine are glossed over andunreferenced. It was therefore impossible to build a fully functional model with just the information providedin [76] and its references.

Several attempts were made, however, to fill in the gaps of missing information, using results from otherworks, literature research, and simple scaling of parameters to fit the reference. This section will thus providea list of the attempts made to fill two major gaps of information:

• Spacing and type of the lattice elements: how are the lattice elements placed and connected in themodel space, and what do they physically represent?

• Charge of vacancies and ions: how is the charge of vacancies and ions determined?

5.4. Validating the full model 69

(a) Extracting a spacing of ≈ 3Å from the horizontal resolutionof Figure 4h in [76].

(b) Extracting the lattice element types of vacancies, ions andoxide from Figure 4c in [76].

Figure 5.10: Extracting lattice element spacing and type from figures in the reference work. Copied and adapted from [76].

Spacing and type of the lattice elements: attempted solutionsTwo aspects of the model lattice are important for understanding the operation of the model:

• Spacing: the position of lattice elements within the model space. This could be a regular grid-like struc-ture, but also a uniform distribution, or a crystal structure. The spacing also influences the connectionsbetween the lattice elements, which in turn may drastically change the implementation of the Poissonsolver.

• Type: what a lattice element represents. Element types can be high-level (oxide, vacancy, interstitialion) or of a lower level (Hf/O atom, Hf+/O− ion).

Neither are clearly defined in the reference model. Its figures, however, do appear to show an indicationof grid element space and types, specifically Figure 4 [76]. Figure 5.10a shows how the spacing was assumedfrom the resolution of one of the figures of the reference model. Figure 5.10b shows how the lattice elementtypes were assumed from the visualization of the model grid. From these figures, the following assumptionswere made:

• Assumed spacing: the assumed spacing is a regular grid with a mutual distance of 3Å.

• Assumed type: the assumed element types is a high-level list of possible states (see alsolatticeElementin Section 4.2).

Since this did evidently not produce usable results, two attempts were made to fit this work’s model tothe reference model: the addition of an interstitial grid, and the generalizing of the lattice coordinates to amonoclinic grid.

• Additional interstitial grid. In the first attempt, oxygen ions would replace oxide elements as the resultof generation and drift events. This seemed unrealistic, as the neighboring oxide element - that wasreplaced by the ion element - would no longer be able to generate a new vacancy-ion pair. In realisticoperation, oxygen ions on generation drift not to another point in the lattice, but rather to a positionbetween the lattice points, an interstitial position (see also Section 4.4) [10, 33, 71]. Therefore a new im-plementation was made, which can be found in the branch interstitial on the complete repository[47].

– The interstitial elements are stored at the end of the list of latticeElement objects. Before im-plementing the interstitial grid, the latticeElement list was stored in MATLAB as a 3D array.After implementing the grid is turned into a regular vector, with all geometric information such asposition and connections stored inside the latticeElements.

– Generation event rate calculation no longer uses the Cartesian components of the local electricfield (Fx , Fy , Fz ) but rather the components that point to the 8 surrounding interstitial positions.


Figure 5.11: Example of a layout of the regular grid, with an additional interstitial regular grid. A single interstitial element exists betweenthe elements of the regular grid.

To achieve this, the 8 unit vectors pointing along the diagonals towards the interstitial positionsdefined by the 8×3 vector set:

U = [u1, u2, · · · , u8]T (5.14)

Multiplying this vector set with the local electric field vector F (x, y, z) then produces the 8 × 1vector of the local electric field projected onto the 8 diagonals:

U ×F (x, y, z) = Fpr o j (5.15)

The values in Fpr o j are then used to calculate the event rates of 8 different events that representthe 8 different interstitial spots where an oxygen ion could be generated.

– The electric potential and temperature profiles within the interstitial grid can still be calculatedusing the regular FDM Poisson solver, but interaction between the regular grid and the interstitialgrid requires unique entries in the Laplacian matrix (see also Section 4.3) since the diagonal edgesconnecting the interstitial elements and the oxide elements are irregular.

Therefore, the entries of the Laplacian matrix at interstitial elements and the directly neighbor-ing oxide elements are modified according to the connections in the 3D graph: the diagonal atan interstitial element is set to its degree (8) and the diagonals at neighboring elements are in-creased by 1 for every neighboring interstitial, as the degree increases. All off-diagonal entries ofthe Laplacian that represent the edges between interstitials and regular grids are set to -1.

In this way, the influence of charge and power in the regular grid is communicated to the intersti-tial grid, and vice versa.

Results: even though the interstitial grid inadvertently both generalized the model and increased itscomplexity, this implementation still did not exhibit the behavior as reported in the reference model.The problem of high local electric fields (see also Section 5.3) was made worse by the relatively smallerdistance between the ions and the oxide, causing rampant generation of vacancies around the ions,which further prevented them from drifting away. It seems that this more physical interpretation oflattice element spacing is either too complex for its purpose, or not realistic enough.

• Adapting the regular grid to a monoclinic structure. After an inquiry session with two of the originalco-authors of the model (Gennadi Bersuker and Luca Larcher) it was clarified that the material struc-ture of the grid plays an important role in the calculation of the state change. Therefore the regular gridwas adapted to a more general structure that represents the monoclinic crystal structure of substochio-metric hafnium oxide [46] (HfOx ), the result of which is displayed in Figure 5.12. This again required anoverhaul of the model implementation:


Figure 5.12: Example of a layout of the monoclinic grid, showing the values for the electric potential φ in a 3D block plot. The grid isdistorted from the regular grid to better represent the crystal structure of hafnium oxide.

– The x,y ,z coordinates of the latticeElements are no longer trivial and instead are calculatedbased on four values that describe a parallellepiped unit of its most stable monoclinic structure:a = 5.136Å, a = 5.136Å, a = 5.193Å, c = 5.371Å and γ= 99.63° [12].

– The calculation of the Laplacian matrix for the Poisson solver requires a complete overhaul: sincenone of the points are regularly spaced now, FDM no longer suffices. Instead, the Finite ElementMethod (FEM) is implemented [2] which can produce a Laplacian matrix for any arbitrary spacedset of points, if they are partitioned in tetrahedral elements (Note: “elements” here refers to theelements used in FEM, not of the lattice) defined by 4 points in the lattice. Figure 5.13 shows thedifference between FDM and FEM.

The subdivision of the monoclinic parallellepiped units in tetrahedral elements is trivial: any 3Dregion with 8 vertices can be filled with exactly 5 tetrahedra. In this way, the irregular lattice waspartitioned into several tens of thousands of tetrahedral elements consisting of 4latticeElementindices. With help of this subdivision, a Laplacian matrix can be constructed.

Both for brevity, and because this implementation did not cause the model to function any better,the full explanation for implementing the FEM Poisson solver will be omitted. A description of anexisting 3D implementation can be found in [2].

Results: just like the previous attempt, this implementation mostly provided further generalizationof the model (by allowing arbitrary coordinates for the lattice elements). Unfortunately, adapting thecrystal structure to a monoclinic structure did not significantly change the functioning of the model, asthe same problems with the kMC engine still persisted.

The two above implementations - as well as the combination of both - were found not to significantly


Figure 5.13: The difference between FDM and FEM, in a 2D case: FDM (left) uses the difference between two data points to determinethe Laplacian matrix. FEM (right) uses triangular (tetrahedral in 3D) elements to determine the Laplacian matrix.

improve the functioning of this work’s RRAM model. For the sake of simplicity, these implementations werereverted to reinstate the initial, more simple implementation, which was explained in Chapter 4.

Spacing and type of lattice elements: potential solutionsThere is, however, still a possibility that the reported simple implementation is not representative of the refer-ence model, which might be the cause of the problems listed in this chapter. The following list provides pro-posals concerning the spacing and type of lattice elements which can potentially cause the model to functionproperly, but were not yet attempted due to a lack of time. These are: using completely continuous coordi-nates, applying an atomistic structure, and neglecting oxygen ions.

• Continuous coordinates. Instead of initializing the full lattice grid as a structure of elements, only theelements that are relevant to the operation of the device (vacancies, ions, electrode-oxide interface)could be generated. This sparsified grid is no longer be bound to a defined structure, as is realisticallythe case within the amorphous region of an oxide grain boundary [90] (see also: Section 4.4). Thisimplementation faces a couple of challenges:

– For the existing elements, their arbitrary coordinates require the FEM implementation of the Pois-son solver, raising the question how to partition the lattice into tetrahedral elements for any gen-eral distribution.

– For the newly created elements, the electric field and temperature profiles should be defined con-tinuously, as new elements are now allowed to generate anywhere in the model space. This posesanother challenge for the implementation of the Poisson solver.

– The calculation of event rates within the existing kMC engine becomes problematic: Monte Carlomethods can only work if there is a finite amount of state changes. If the electric field and tem-perature profiles are defined continuously, there are by definition an infinite amount of events.

• Atomistic structure. Define the lattice grid as a discrete set of points, but modify the types of the latticeelements to represent the low-level atomistic structure of hafnium oxide. The coordinates of the struc-ture are still randomized within a continuous space, but oxygen vacancies can now only generate onpre-defined oxygen atom positions. A form of this implementation is seen in [90]. This implementationis also challenging:

– The initial positions and connections of the atoms, in both the amorphous region in the GB andthe crystalline zone in the grain, can not be randomized but must be based on knowledge aboutthe nano-scale structure of hafnium oxide.

– Interstitial positions can possibly become vague within the amorphous region: because the atomsare unorganized, the coordinates of the position between the elements can not be trivially de-duced.

– Similarly, the drift directions of an oxygen ion to other interstitial positions becomes hard to de-fine, and the variable distance of the jump also starts to influence the event rates.


• Neglect interstitial oxygen ions. Simpler versions of OxRAM models do not consider the drifting ofoxygen ions but instead the drifting of vacancies itself [1, 43, 82]. The generation and recombinationevents do not respectively create or destroy an vacancy-ion pair, but only a vacancy. This implementa-tion would require relatively little adaption:

– The drift rate calculation for vacancies needs to be adapted, since the reference model assumedthat vacancies never drift [76]. The drift rate calculation for ions - and the 'O−2' type of element- must be removed.

While this implementation is simpler and might cause the model to function correctly, it eliminates theadvantage of physical accuracy that was accounted for in the reference model.

The above proposals are neither proven nor disproven and may serve as pointers for future work. Thespacing and type of lattice elements is not the only information missing though: the reference also does notclearly explain the charge of vacancies and ions.

Charge of vacancies and ions: attempted solutionsThe charge of vacancies and ions plays a core role in the determining of the electric field. Padovani et al.[76] mention that, by taking the charge of these lattice elements into account while calculating the electricfield profile, Coulomb repulsion and attraction are automatically accounted for. Therefore, using the propervalues for the charge of vacancies and ions is crucial for proper operation of the kMC engine. Unfortunately,the strategy of calculating these charges is not explained in the reference work.

Once again, several attempts were made to fill in this gap of information. In short, these attempts are:

• Constant charges for both vacancies and ions.

• Reducing vacancy and ion charges by a constant.

• Reducing vacancy charge according to the electron flow rate of the paths.

• Reducing vacancy charge according to neighboring vacancies.

• Reducing vacancy charge according to their electron occupation probability.

Figure 5.14: Constant charge lattice elements placed inside the oxide: positively charged V+2 vacancies (blue) and negatively chargedO−2 ions (red).

• Constant charges for both vacancies and ions. The first attempt for determining the charge of va-cancies and ions was a simple assumption based on the symbols used for vacancies (V+2) and oxygenions (O−2) in the reference work. A vacancy was thus assumed to have a charge of +2e, and an ion wasassumed to have a charge of -2e. A visualization of this implementation is displayed in Figure 5.14.

Results: the main problem that arises from this assumption is that the distance between lattice ele-ments is very small (3 Å, see also Section 4.6). Which causes the magnitude of the electric field to bevery large, which causes a runaway generation of vacancies regardless of the applied voltage, as wasexplained earlier in Section 5.3. It can therefore be concluded that this assumption is incorrect.


Figure 5.15: Reduced charge by a constant: positively charged V+2 vacancies (blue) and negatively charged O−2 ions (red). Less colorsaturation implies less charge.

• Reducing vacancy and ion charges by a constant. The simplest way of reducing the extremely largelocal field that exists next to the point charges, is to reduce the charge by a constant, as visualized inFigure 5.15. The charge density ρ was thus multiplied by a constant value kρ < 1, which was considereda fitting parameter. This effectively reduces the non-homogeneous electric potential field caused by thecharges, while not touching the homogeneous field caused by the applied voltage on the electrodes.

Results: Reducing the charges by a constant proved a successful solution to stop the runaway genera-tion of vacancies next to other vacancies or ions. The value of the constant is, however, debatable, fortwo reasons:

1. A constant reduction of charge is an assumption with no physical basis and is therefore difficultto obtain from literature.

2. For low vacancy densities, e.g. before forming, the non-homogeneous electric field caused bycharges is orders of magnitude smaller than for high vacancy densities, e.g. in LRS. Reducingthe constant to account for the large non-homogeneous field for high vacancy densities, meansthat for low vacancy densities the electric field magnitude is too small to cross the generationactivation energy barrier. Vice versa, a larger constant allows generation to occur in low vacancydensities, but does not reduce the large charge-induced electric field for high vacancy densities,which still causes the problem of runaway vacancy generation.

It was therefore observed that, whereas reducing the vacancy and ion charge does solve the voltage-independent runaway vacancy generation problem, it can not be determined by a constant: a distinc-tion must be made between high and low vacancy densities.

Figure 5.16: Percolation path based charge: positively charged V+2 vacancies (blue) and negatively charged O−2 ions (red). Less colorsaturation implies less charge. The charge of the vacancies is determined by the electron flow rate of the percolation path it is in.


• Reducing vacancy charge according to the electron flow rate of the paths. To find a physically-basedsource of charge reduction, the charge transport model was investigated (see also Section 4.4). Thephysical process behind the charge transport involves the capturing and emitting of an electron intoa trap (Trap Assisted Tunneling). A logical conclusion from this would be that the more electrons flowthrough a vacancy, the more it receives negative charge. Research on the formation of vacancies [10,27] confirms this, and even shows that the charge of a vacancy can become negative, due to manyelectrons being captured at once. This also explains how negatively charged oxygen ions can be repelledby vacancies.

The TAT solver framework [59, 91], however, is based on the assumption that no charge build-up occursinside a trap [59] and therefore does not mention any strategy of calculating the charge of a trap thathas captured charge carriers. Determining the vacancy charge is therefore another gap of informationthat needs to be filled in.

This was attempted by relating the charge of a trap, or vacancy, to the electron flow rate of the perco-lation path it exists in, as displayed in Figure 5.16. The electron flow rate of the percolation path Rpath

represents the amount of electrons flowing through the trap per second and is already calculated withinthe TAT solver to determine the current contribution of a single path. The charge of a trap in this pathcould therefore be determined by a linear equation:

qi =+2e−kq ·Rpath (5.16)

where kq will be used as a fitting parameter, and qi is limited to -2e.

Results: while this implementation does partly solve the problem of the previous implementation re-garding the magnitude of the local electric field, it only does so when enough current flows through thedevice. Furthermore, even though a negative charge pushes ions away from the vacancy, the electricfield magnitude is still very large, but only flipped. The consequence of this is that vacancies still gen-erate rapidly around existing vacancies regardless of the applied voltage. This assumption is thereforeassumed incorrect.

Figure 5.17: Metallization charge: positively charged V+2 vacancies (blue) and negatively charged O−2 ions (red). Less color saturationimplies less charge. The charge of vacancies is determined by the amount of vacancies in its vicinity.

• Reducing vacancy charge according to neighboring vacancies. Another physically-based source ofvacancy charge reduction is metallization. Microscopic models have shown that on a structural level,oxygen vacancies between hafnium atoms lose their charge once they cluster together [27, 109]. Thisrepresents the formation of a metallic, substochiometric region HfOx , meaning that there are less than2 oxygen atoms per 1 hafnium atom. In a structural sense, this means that for large vacancy densities,hafnium atoms rather connect to each other than leave holes in the lattice. This causes a reduction incharge as there are less disconnected hafnium atoms.

The effect of metallization may be modelled by a simple linear function of the distance from a vacancy,which reduces the charge of the vacancies around it. The effect is displayed in Figure 5.17. For everyvacancy element i , all surrounding vacancy elements j (within a distance limit ri j < rmax ) multiplytheir charge as:


q j = q j ·(1−km

(1− ri j

rmax

))(5.17)

where q j is the charge of element j , and km ∈ [0,1] is a factor representing the strength of metallization.The higher km , the more vacancy density influences charge. Therefore, it is a fitting parameter.

Results: for higher vacancy densities, the non-homogeneous electric field caused by the large clus-ter of charges is succesfully reduced by applying Equation 5.17. For lower vacancy densities however,metallization can not reduce charge and the electric field magnitude is still too large. Combining themetallization implementation with a (relatively low) constant charge reduction implementation how-ever, also keeps the electric field magnitude for low vacancies within acceptable bounds.

There is however still a problem: large clusters of oxygen ions can also form in the model, similarlycausing a very high local electric field magnitued. Reducing their charge according to their density toprevent this is not represented by any physical process, as oxygen ions exist separately in interstitialpositions and are not inclined to connect. Another solution must therefore be found to reduce thecharge of the ions in the model.

Figure 5.18: Electron occupation probability based charge: positively charged V+2 vacancies (blue) and negatively charged O−2 ions(red). Less color saturation implies less charge. The charge of vacancies is determined by the probability of an electron existing insidethe vacancy.

• Reducing vacancy charge according to their electron occupation probability. The final implementa-tion attempt was based on equations from Larcher [59], which form the basis of calculating the electroncapture and emission rates per trap for the TAT solver (see also Section 4.4). These equations involvethe probability that an electron exists in a trap, the electron occupation probability:

ft ,i = τe,i ·(τc,i +τe,i

)−1 (5.18)

where τc,i and τe,i are the electron capture and emission times, respectively. ft ,i represents a probabil-ity, which means that ft ,i ∈ [0,1].

Note that this probability is different from the Fermi-Dirac occupation probability fi (E), which showsthe probability of an electron existing on a certain energy level E : rather, ft ,i shows the probability ofan electron existing inside the vacancy.

The capture and emission times are already calculated in the TAT solver framework for every trap (orvacancy) in the lattice and therefore the electron occupation probability for every trap can be deducedusing Equation 5.18. The charge of a vacancy may be directly linked to the electron occupation proba-bility by:

qi = q0 −q · ft ,i (5.19)

where qi is the charge of vacancy i , q0 is the unoccupied charge of vacancy i and −q is the charge of anelectron. The effect of this is visualized in Figure 5.18.


The problem of Equation 5.19 is that it is still unknown what the charge is of an unoccupied vacancy(q0). For this, again, the results of microscopic models [27] was used to find that the type of vacancythat is most prone to clustering is that of a single positive charge: V+. It was therefore assumed thatunoccupied vacancies have a charge of q0 =+1e and the charge of vacancy i was thus determined as:

qi = q · (1− ft ,i)

(5.20)

Results: even though this implementation is mostly based on the references, it doesn’t always reducethe charge of vacancies enough. The electron occupation probability can be anywhere between 0 and1, and thus vacancies can have any charge between +0e and +1e, which may still cause a local electricfield that is too large. Furthermore, not all oxygen vacancies inside the lattice have a charge of +1e [27]:possible charges include any option from -2e to +2e. The assumption that q0 = +1e is therefore notcompletely correct. Furthermore, oxygen ions are not considered by the TAT solver. These still havea debatable charge which can cause too large electric field magnitudes regardless of electrons flowingthrough the vacancies.

Of the above list, all implementations were attempted, but eventually the final implementation producedthe most realistic results. In the future, combinations of these attempts may be considered, or another, newway of determining the charge of vacancies and ions. Vacancy and ion charge appears to be an importantaspect of RRAM switching, since many processes are field-driven. Therefore it is peculiar how little researchexists on this subject.

In conclusion, there are still too many issues with the full model for it to accurate reproduce the electricalcharacteristics of an RRAM device. However, pointers were provided in this section to improve the model sothat in the future it may be used to more accurately investigate the effect of defects on OxRAM forming andswitching.

For this work, only the Poisson solver and the TAT solver were validated. Therefore, even though the time-dependent processes of the reference model can not be reproduced, the charge transport model can still beused to investigate the effects of defects on the resistance of steady-state RRAM. The following chapter willthus use the Poisson solver, the TAT solver and the defect injector to research the effect of RRAM defects inLRS and HRS.

6Defect analysis

In this chapter, the model described in Chapter 4 and partially validated in Chapter 5 will be applied to in-vestigate the effects of defects on the conductivity of RRAM pre-forming and in LRS. First, the general exper-imental setup will be presented. Then the four defect types listed in Section 2.4 are injected into the latticegrid: vacancy density variations due to the presence or absence of grain boundaries, oxide thickness varia-tion, electrode roughness, and impurities that contaminate the device.

6.1. Experimental setupFor every defect, a range of values is chosen defining a property of the defect, and a voltage sweep is per-formed from 0 to 3V, while calculating the current using the charge transport model described in Section 4.4.To account for statistical fluctuation and outliers, Nsi m = 100 simulations will be performed for every ex-periment. All graphs show the mean of all simulations, together with a shading implying the range of thesimulation results. The dimensions and parameters of a nominal device are listed in Table 6.1.

Table 6.1: A list of the parameters describing the nominal, defect-free device.

Parameter Description Nominal value

Nx Total x-dimension of model space. 30 elements (9nm)Ny Total y-dimension of model space. 30 elements (9nm)Nx Total z-dimension of model space. 40 elements (12nm)Nel Part of z-dimension representing electrodes. 4 elements (1.2nm)NV Vacancy density outside grain boundary. 3.0×1019 cm−3

NV ,GB Vacancy density inside grain boundary. 2.1×1021 cm−3

rGB Radius of grain boundary. 4nmTamb Ambient temperature 300K

The rest of all properties, together with physical constants that do not change between simulations, canbe found in the full properties list in Appendix A.

For experiments in LRS, a filament had to be inserted into the oxide manually, because the kMC enginegoverning the state change of the model was unable to function correctly (see also Section 5.3 and Sec-tion 5.4). This means that the filament shape must also be pre-determined manually. 3D images of conduc-tive filaments in OxRAM were shown to have an hourglass shape [15], as displayed in Figure 6.1a. Therefore,the shape of the filament was described by an hourglass-shaped region:

dc (i , j ) < r (k) (6.1)

where dc (i , j ) is the index distance from the center projected on a plane parallel to the electrodes, andr (k) is the index radius of the filament according to:

dc (i , j ) =√

(i −Nx /2)2 + (j −Ny /2

)2 (6.2)

79

80 6. Defect analysis

r (k) = 0.04 · (k −Nz /2)2 +4 (6.3)

where i , j and k are the element indices in respectively x, y and z directions, and Nx , Ny and Nz arethe amount of elements in those directions. Within this region, the probability of a lattice element being avacancy (PV ∈ [0,1]) is determined by:

PV (i , j ,k) = 1−0.6 · dc (i , j )

r (k)(6.4)

The hourglass-shaped filament resulting from Equation 6.1 and Equation 6.4 is displayed in Figure 6.1b.

(a) Visualization of the shape of a conductive filament in RRAM, ob-tained by 3D imaging. Adapted from [15].

(b) A lattice grid modeling an RRAM device in LRS. with a manuallyinserted conductive filament shape determined by Equation 6.1 andEquation 6.4.

Figure 6.1: Manually inserting a conductive filament in the oxide of the lattice grid.

The experimental setup for each defect is described below.

Setup for vacancy density fluctuationsThe main cause for unexpected vacancy density fluctuations is the presence or absence of grain boundaries(GBs) [9, 78], therefore, to investigate the effect of vacancy density fluctuations, the experimental setup inSection 6.1 was modified to remove the GB in the middle of the oxide. Two examples of the lattice grid withand without the GB are displayed in Figure 6.2.

(a) With a GB. (b) Without a GB.

Figure 6.2: Examples of the lattice grid, generated by the lattice builder described in Section 4.6, with and without a GB.

6.2. Results 81

Oxide thickness variationTo investigate the effects of oxide thickness variation, the total z dimension of the model space (Nz ) wasreduced and increased. The general experimental setup from Section 6.1 was used, with Nz taking the values:

• Nz = 20 (tox = 4.8nm);

• Nz = 30 (tox = 7.8nm);

• Nz = 40 (tox = 10.8nm);

• Nz = 50 (tox = 13.8nm).

Electrode roughnessThe effects of electrode roughness were investigated by appending a uniformly random amount of electrodeelements to the electrode-oxide interfaces (see also Section 4.6). Experiments were performed for four dif-ferent values of roughness, defined by the maximum amount of electrode elements which can randomly beappended:

• roughness = 1 (σel = 0.3nm);



• roughness = 4 (σel = 1.2nm).

ImpuritiesImpurities are contaminations of the oxide by a foreign material and can have many different shapes andsizes. For the purpose of researching their effect, two types of impurities are injected into the lattice grid: ablocking impurity and a conducting impurity. The blocking impurity is represented by an air bubble in themiddle of the oxide, obstructing both the grain boundary and the conductive filament. The conducting im-purity is represented by a bump in the lower electrode, locally reducing the oxide thickness. These impuritiesare described by four parameters: type, width, height, and position (see also Section 4.6). The parameters arelisted in Table 6.2. The resulting impurity defects are displayed in Figure 6.3.

Parameter Blocking impurity Conducting impurity

Type 'air' 'lower' (electrode)Width 4.0nm 3.0nmHeight 3.0nm 8.0nmPosition [4.5nm,4.5nm,8.0nm] [3.0nm,6.0nm,0.0nm]

Table 6.2: The parameters used in the injection of the blocking and conducting impurities.

6.2. ResultsIn this section, the results of the experimental setup in Section 6.1 are presented for the four selected defecttypes: vacancy density fluctuations, oxide thickness variation, electrode roughness and impurities.

Vacancy density fluctuationsThe results of the simulations analysing the effect of GBs affecting the vacancy density are displayed in Fig-ure 6.4.

Figure 6.4a shows that the presence or absence of a GB does have an effect on the RRAM device in pre-forming conditions. Without a GB, the device has a larger resistance, especially for lower voltages. The effecton conductivity does not appear to be significant for higher voltages, apart from reducing the variation ofconductivity, as displayed by the red and blue shaded areas. This is probably due to the GB acting as a priorityconductive path, which guides the percolation paths of the electrons from the cathode to the anode (see alsoSection 4.4).


(a) Blocking air bubble in an unformed device. (b) Blocking air bubble in LRS.

(c) Conducting electrode bump in an unformed device. (d) Conducting electrode bump in LRS.

Figure 6.3: Four examples of lattice grids with injected impurities, generated by the implementation described in Section 4.6 based onthe parameters listed in Table 6.2.

(a) In an unformed device. (b) In LRS.

Figure 6.4: The current through the device for voltages from 0 to 3V, in the case of a GB existing in the oxide (blue) and no GB existing inthe oxide (red). The range of the simulation results are displayed by shadings in the respective colors. Since the effect of GBs on LRS cannot be investigated by this model, subfigure (b) shows just the defect-free conductivity in LRS (blue).

What is not displayed in Figure 6.4a is the effect that a GB has on the generation of new vacancies. Previousmodels and measurements have shown that conductive filaments favor GBs to form [9, 10, 76, 90]. Becausethis model’s kMC engine is unable to function correctly (see also Section 4.5 and Section 5.3), this can not

6.2. Results 83

be verified by this model, but it is clear from previous knowledge the effect of GB outside of its conductiveproperties is significant

Figure 6.4b shows the conductivity of the device described in Section 6.1 with the manually inserted con-ductive filament, displayed in Figure 6.1 and described in Equation 6.1 and Equation 6.4. It appears that theconductivity of the filament is only slightly larger than that of the unformed device. This might be due to theoxide being too thick (10.8nm) to function properly as an RRAM filament. For higher voltages (around 3V),the unformed device appears to conduct better than the formed device. The exact reason for this is unknown,but might be an error in the implementation of the charge transport model (Section 4.4 and Section 5.2).

Another possible reason for the perceived loss of conductivity in LRS is that the charge transport model isnot able to realistically model the defect subband that is created for a sufficiently high density of vacancies.The reference model states that this is possible using the same TAT framework as explained in Section 4.4[59, 76, 91]: as the mutual distance between the traps decreases, their capture radii overlap, reducing thepotential barrier and increasing the tunneling probability to 1. From these results, this barrier reducing effectappears to be insufficient to model the drift current. Instead, a different charge transport model may have tobe implemented for the device in LRS, as the drift current is not modeled correctly by the TAT solver. Othermacroscopic models already do this [1, 43, 82].

Oxide thickness variationThe results of the four simulation sweeps in an unformed device and in LRS are displayed in Figure 6.5.


Figure 6.5: The current through the device for voltages from 0 to 3V, for four different oxide thicknesses. The means of each set of 100simulations are displayed by lines, and the range of all simulation results is displayed by shadings in the corresponding colors.

In an unformed device, the oxide thickness has a clear effect on the conductivity, as shown in Figure 6.5a.For thin oxides, a variation of a few nanometer can already modify the current by five orders of magnitude,as can be seen from the results of tox = 7.8nm (green) and tox = 4.8nm (red). The oxide thickness has aninsignificant effect on the variation of the device conductivity, because the statistical spread of the currentdoes not change as the thickness varies. Furthermore, a thinner oxide also increases the electric field insidethe device, subsequently increasing the rate of vacancy generation. This can cause the filament to form earlierthan expected.

Figure 6.5b shows the effect of varying oxide thickness on a device in LRS. Once again, it appears that thecharge transport model can not successfully reproduce LRS drift current. However, the current does visiblyincrease for all thicknesses. The effect of oxide thickness variation is less significant than in an unformeddevice, most likely because the majority of the current is caused by the conductive filament instead of directtunneling.

Electrode roughnessThe results of the four simulation sweeps in an unformed device and in LRS are displayed in Figure 6.6.



Figure 6.6: The current through the device for voltages from 0 to 3V, for four different values of electrode roughness. The means of eachset of 100 simulations are displayed by lines, and the minimum and maximum values are displayed by triangles in the correspondingcolors.

Figure 6.6a shows that in an unformed device, electrode roughness has a relatively small effect on theconductivity. A higher roughness causes a slightly larger current, because the randomly appended electronelements locally decrease the oxide thickness. The ends of the appended electron elements also cause a localpeak in the electric field, as is displayed in Figure 6.7. These local peaks will accelerate forming, which cancause an uncontrolled hard dielectric breakdown to occur [32, 79].

Figure 6.6b shows that the current increases only slightly after manually inserting a conductive filament,due to the issues posed earlier. For roughnesses of 1, 2 and 3 elements however, the variation of currentvisibly reduces, indicating that the local variations of oxide thickness no longer have a dominant effect onthe conductivity as the percolation paths prefer the filament. A roughness of 4 elements shows a very widespread and a very high mean current. The reason for this is unknown, but most likely caused by an error inthe charge transport model, which may require further research.

ImpuritiesFigure 6.8 shows the results of measuring the current through the RRAM device when the two types of impu-rities are injected, for an unformed device and in LRS.

Figure 6.8a shows that the conducting impurity greatly increases the conductivity of the device, as was ex-pected. The effect is much more significant than that of the blocking impurity, mostly because the resistanceof the device is already quite large before the device is formed. Another significant effect of the conductingimpurity is not directly related to the device current, but to the electric field it causes, as is displayed in Fig-ure 6.9. The magnitude of the electric field at the tip of the bump is significantly higher than in the rest ofthe oxide, which causes the forming of the conductive filament to prefer the impurity. The high field mag-nitude will also speed up generation of vacancies, causing unreliable forming and possibly a hard dielectricbreakdown.

Figure 6.8b shows, once again, that the charge transport model is not able to successfully reproduce thedrift current of an RRAM device in LRS. It does show, however, that the impurities have an effect on the spreadof the different simulations. Nominally (green) the spread of current decreases in LRS, due to percolationpaths favoring the conductive filament. In the case of the conducting impurity, the current variation changeslittle, because the majority of current is caused by the locally thin oxide at the electrode bump. In the case ofthe blocking impurity, the variation does decrease, because here the majority of current is still caused by theconductive filament.

This concludes the analysis of defects in an RRAM device, using the charge transport model presentedin Section 4.4 and the experimental setup Section 6.1. In conclusion, the effects of these defects are clearlyvisible and unique, but may require further validation.

6.2. Results 85

Figure 6.7: Examples of the four electrode roughness values producing the results displayed in Figure 6.6. The left four figures show thelayout of the lattice grid for the indicated electrode roughness. Electrodes are displayed as triangles, and vacancies as blue circles. Theright four figures show the respective electric fields, excluding the contribution of vacancies.



Figure 6.8: The current through the device for voltages from 0 to 3V, for two types of impurities. The means of each set of 100 simulationsare displayed by lines, and the range of all simulation results is displayed by shadings in the corresponding colors.

Figure 6.9: The electric field in a lattice grid with a conducting impurity, excluding the electric field caused by vacancies.

7Discussion

In this chapter, the results of this work are critically observed, but mostly the process of getting these results isdiscussed. This is because, even though the primary focus of this work was to elaborate an existing physicalmodel with the ability to simulate defects, it eventually became more of an example of the importance ofreproducability of scientific publications.

Unexpected work loadOriginally this work was planned to focus more on the inclusion of defects instead of the physical operation ofdefect-free RRAM, since the latter has already been extensively researched, as described in Chapter 3. Thus,a physical, macroscopic model had to be selected which appeared to have been documented extensivelyenough to rebuild it with relative ease, after which the main focus could turn to the defects. The comparisonof existing macroscopic models revealed that the work by Padovani et al. [76] was the best candidate for thisdefect-free reference model.

However, after several months of attempting to recreate the model, it seemed that it requires an unex-pectedly enormous effort to understand all of the concepts and theories involved in the physical processesof RRAM operation, especially for a CE master student and outsider. Even though many of the concepts inthe reference work were properly sourced by references and explained, successfully recreating the model stillrequired specific knowledge about quantum physics, solid-state physics, material science, electromagnetics,applied mathematics, and computer science. Obtaining this knowledge and effectively applying it may havebeen too ambitious for a master thesis.

Reproducability issuesMany of these fields only had to be explored and understood because of missing information, assumptionsand unclear definitions in the reference work. The most troublesome are the parameters of the kMC engine(see also Section 5.3), the implementation details of the lattice grid (Section 5.4), and the charge value of thevacancies and ions (see alsoSection 5.4). As was explained in Chapter 5, even with the provided informationand knowledge on the relevant subjects, it was still not possible to reproduce the results that were publishedin the reference work.

Therefore, apart from providing an in-depth analysis of the physical processes used in the referencemodel, this thesis also raises questions about its reproducability. A scientific research has little value if itis not reproducable. It is difficult to judge the reproducability of any scientific work: although the physicalparts of the reference model were clearly documented and sourced, the specific details about the implemen-tation were either assumed or obfuscated. Either way, this work has shown that without these details, thereference model is too complex to reproduce correctly.

Attempts to fill in informationStill, however, several attempts were made to fill in the gaps that were left by this missing information. Theentire MATLAB structure of the model, connecting the physical modules described in the reference in an

87

88 7. Discussion

efficient way, had to be constructed from scratch (see also Section 4.2). The WKB tunneling probability cal-culator (Section 4.4), the Tsu-Esaki direct tunneling calculator (Section 4.4), and the Poisson equation solver(Section 4.3) are examples of core parts of the reference model which were mentioned without a referenceand were also constructed from scratch.

Value of this workThe results of this work, presented in Chapter 5 and Chapter 6, have been shown and described to not be com-pletely realistic. The reasons behind these errors has been explained and a solution hypothesized. Therefore,this work may not be a viable reference for the construction of a more compact defect model, as was its mo-tivation (see also Chapter 1). Instead it may provide a basis and source of knowledge for an improvementthat may produce more realistic results. To aid this, the information missing in the reference work has beenextensively explained in Chapter 4 and its components were both validated and invalidated in Chapter 5.

Future workThus, in the future, this model could be adapted and improved to be able to successfully model the full resis-tive switching process of RRAM, as well as the effects of defects on its operation. The source code is providedpublicly [47] and the explanation of its implementation can be found in this work. The model is already ableto inject defects into a lattice grid and calculate the current through a RRAM device in HRS. The irregularitiesof defects are taken into account by the combination of element-based electrodes and custom-made Poissonequation solver. As such, this work will serve a basis for further improvement of the model, so that it caneventually be used for its intended purpose, which is to simulate the full forming and switching process of asingle RRAM device, on a physical level, with the inclusion of defects.

The power of this model is its versatility: because of its physical nature, any shape and size of defect canbe injected into the lattice grid and its effects observed. If it could be made to work correctly and verified,new types of defects could be added to it next to the types discussed in this model Section 2.4. Eventually, theresults of the model can be compacted into a SPICE model, and tests can be developed for any type of defect.

8Conclusion

This work has laid the basis for a macroscopic, physical-based RRAM defect model, which in the future canbe used for advanced, in-depth defect analysis. It has also provided an extensive explanation of all physicalprocesses associated with RRAM operations.

Motivated by the lack of physical basis in RRAM defect modeling, this work aimed to connect the physicalprocesses that drive the generation and recombination of a conductive filament in the RRAM device, to itselectrical characteristic of resistive switching. Background information was provided about the function ofmemory, emerging non-volatile memory technologies and the operating principles of RRAM devices. Also,the manufacturing process of RRAM devices was investigated, and its most common defects were described.

After providing the necessary background information to understand the principles of RRAM defects, thestate of the art of RRAM defect modeling was explored in more detail. With the goal of selecting a defect-free model to elaborate with the inclusion of defects, existing defect-free models were compared accordingto their usability. The macroscopic physics-based model by Padovani et al. was chosen to be used as thereference model.

The structure of the reference model was divided in general modules and three physical modules: thePoisson equation solver, the charge transport model and the Monte Carlo state change engine. The noveladdition to the model is the defect injector, which modifies the initial state of the model to include one ormore of four defect types: vacancy density fluctuations, oxide thickness variation, electrode roughness andimpurities.

The model was then validated, and it was found that missing information and unmentioned assumptionsmade it impossible to reproduce the results of the reference model correctly, in part due to the confusingdescription of the referenced Monte Carlo engine. The charge transport model and Poisson equation solverwere, however, sufficiently validated and could be used to simulate the effect of defects on the conductivityof a steady-state RRAM device. The results of these steady-state measurements showed the effect of defectson the conductivity of the device and its internal electric field. The effects were used to explain the possiblycatastrophic consequences of the defects.

Finally, a collection of all issues concerning the reproducability of the reference model and the validity ofthis work’s results was presented in a discussion. This work was presented as a basis for further improvementof the model in the future. It could eventually be used for the development of defect characterization and testdevelopment, based on low-level physical simulations of any arbitrary type of defect.

89

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AModel default properties

The following table is a complete overview of all physical properties used in the simulations of the model.Note that variable parameters, such as device dimensions or ambient temperature, are not mentioned herebecause these variables are directly related to specific simulations and change between them. These variableswill therefore be mentioned where they are relevant.

Property Value DescriptionGeneral properties

εr 21 Relative dielectric constant of HfO2

kth 1.5K/Wm Thermal conductivity of HfO2

Lattice element properties∆x 3Å Lattice spacing step

EC ,el 4.45eV Electrode conduction band energyEC ,i l 2.9eV Interface layer conduction band energyET,µ 1.9eV Trap energy mean∆ET 0.5eV Trap energy deviation

rt 5.64Å Trap capture radiusTAT properties

me 0.18ħω0,ox 0.07eV Effective phonon energy in HfO2

ħω0,i l 0.06eV Effective phonon energy in interface layerφg 0,el 0eV Band gap energy of electrodeφg 0,ox 5.8eV Band gap energy of HfO2

φg 0,i l 8.9eV Band gap energy of interface layerEF,el 0 Relative location of Fermi energy in gap from conduction band in electrodeEF,ox 1/2 Relative location of Fermi energy in gap from conduction band in oxideEF,i l 1/4 Relative location of Fermi energy in gap from conduction band in interface layer

S 17 Huang-Rhys factor in HfO2

kMC propertiesEa 2.9eV O-Hf bond breakage zero-field effective activation energy

Ea,d 1.5eV Diffusion activation energyEa,r 0.2eV Recombination activation energy

p0 5.2eÅ O-Hf dipole momentν 7×1013 Hz Effective vibration frequency of O-Hf bonds

kD 3eÅ Factor dependent on molecular properties of HfO2

99

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Modeling the physics of RRAM defects

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